Chapter 6: Linear Momentum and Collisions

Problems & Exercises

6.1 Linear Momentum and Force

  1. (a) Calculate the momentum of a 2000-kg elephant charging a hunter at a speed of [latex]7\text{.}\text{50 m/s}[/latex].
    (b) Compare the elephant’s momentum with the momentum of a 0.0400-kg tranquilizer dart fired at a speed of [latex]\text{600 m/s}[/latex].
    (c) What is the momentum of the 90.0-kg hunter running at [latex]7\text{.}\text{40 m/s}[/latex] after missing the elephant?
    Solution: (a) [latex]\text{1.50}×{\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{kg}\cdot \text{m/s}[/latex]; (b) 625 to 1; (c) [latex]6\text{.}\text{66}×{\text{10}}^{2}\phantom{\rule{0.25em}{0ex}}\text{kg}\cdot \text{m/s}[/latex]
  2. (a) What is the mass of a large ship that has a momentum of [latex]1\text{.}\text{60}×{\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}\text{kg}·\text{m/s}[/latex], when the ship is moving at a speed of [latex]\text{48.0 km/h?}[/latex]
    (b) Compare the ship’s momentum to the momentum of a 1100-kg artillery shell fired at a speed of [latex]\text{1200 m/s}[/latex].
  3. (a) At what speed would a [latex]2\text{.}\text{00}×{\text{10}}^{4}\text{-kg}[/latex] airplane have to fly to have a momentum of [latex]1\text{.}\text{60}×{\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}\text{kg}·\text{m/s}[/latex] (the same as the ship’s momentum in the problem above)?
    (b) What is the plane’s momentum when it is taking off at a speed of [latex]\text{60.0 m/s}[/latex]?
    (c) If the ship is an aircraft carrier that launches these airplanes with a catapult, discuss the implications of your answer to (b) as it relates to recoil effects of the catapult on the ship.
    Solution: (a) [latex]8\text{.}\text{00}×{\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{m/s}[/latex]; (b) [latex]1\text{.}\text{20}×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{kg}·\text{m/s}[/latex]; (c) Because the momentum of the airplane is 3 orders of magnitude smaller than of the ship, the ship will not recoil very much. The recoil would be [latex]-0\text{.}\text{0100 m/s}[/latex], which is probably not noticeable.
  4. (a) What is the momentum of a garbage truck that is [latex]1\text{.}\text{20}×{\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{kg}[/latex] and is moving at [latex]10\text{.}\text{0 m/s}[/latex]? (b) At what speed would an 8.00-kg trash can have the same momentum as the truck?
  5. A runaway train car that has a mass of 15,000 kg travels at a speed of [latex]5\text{.4 m/s}[/latex] down a track. Compute the time required for a force of 1500 N to bring the car to rest.
    Solution: 54 s
  6. The mass of Earth is [latex]5\text{.}\text{972}×{10}^{\text{24}}\phantom{\rule{0.25em}{0ex}}\text{kg}[/latex] and its orbital radius is an average of [latex]1\text{.}\text{496}×{10}^{\text{11}}\phantom{\rule{0.25em}{0ex}}\text{m}[/latex]. Calculate its linear momentum.
  7. *An elephant and a hunter are having a confrontation.
    A drawing of an elephant, on the left, and hunter, on the right. An x y coordinate system is shown, with positive x to the right and positive y up. The elephant is labeled with m E = 2000.0 k g, and vector v E = 7.50 meters per second times I hat. An arrow above the v E vector points to the right. The hunter is labeled with m hunter = 90.0 k g, and vector v hunter = 7.40 meters per second times I hat. An arrow above the v hunter vector points to the right. Between the hunter and elephant is a dart with a long arrow pointing to the left drawn near it and labeled vector v dart = 600 meters per second times minus I hat, and m dart = 0.0400 k g.
    (a) Calculate the momentum of the 2000.0-kg elephant charging the hunter at a speed of 7.50 m/s
    (b) Calculate the ratio of the elephant’s momentum to the momentum of a 0.0400-kg tranquilizer dart fired at a speed of 600 m/s
    (c) What is the momentum of the 90.0-kg hunter running at 7.40 m/s after missing the elephant?
  8. *A skater of mass 40 kg is carrying a box of mass 5 kg. The skater has a speed of 5 m/s with respect to the floor and is gliding without any friction on a smooth surface.
    (a) Find the momentum of the box with respect to the floor.
    (b) Find the momentum of the box with respect to the floor after she puts the box down on the frictionless skating surface.
    Solution: (a) magnitude: 25kg⋅m/s ; 

    (b) same as (a)

  9. *A car of mass 2000 kg is moving with a constant velocity of 10 m/s due east. What is the momentum of the car?
  10. *What is the average momentum of an avalanche that moves a 40-cm-thick layer of snow over an area of 100 m by 500 m over a distance of 1 km down a hill in 5.5 s? Assume a density of 350 kg/m3 for the snow.
    Solution: 1.3×109kg⋅m/s
  11. *What is the average momentum of a 70.0-kg sprinter who runs the 100-m dash in 9.65 s?

6.2 Impulse

  1. A bullet is accelerated down the barrel of a gun by hot gases produced in the combustion of gun powder. What is the average force exerted on a 0.0300-kg bullet to accelerate it to a speed of 600 m/s in a time of 2.00 ms (milliseconds)?
    Solution: [latex]9\text{.}\text{00}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}N[/latex]
  2. Professional Application
    A car moving at 10 m/s crashes into a tree and stops in 0.26 s. Calculate the force the seat belt exerts on a passenger in the car to bring him to a halt. The mass of the passenger is 70 kg.
  3. A person slaps her leg with her hand, bringing her hand to rest in 2.50 milliseconds from an initial speed of 4.00 m/s.
    (a) What is the average force exerted on the leg, taking the effective mass of the hand and forearm to be 1.50 kg?
    (b) Would the force be any different if the woman clapped her hands together at the same speed and brought them to rest in the same time? Explain why or why not.
    Solution: (a) [latex]2\text{.}\text{40}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{N}[/latex] toward the leg; (b) The force on each hand would have the same magnitude as that found in part (a) (but in opposite directions by Newton’s third law) because the change in momentum and the time interval are the same.
  4. Professional Application
    A professional boxer hits his opponent with a 1000-N horizontal blow that lasts for 0.150 s.
    (a) Calculate the impulse imparted by this blow.
    (b) What is the opponent’s final velocity, if his mass is 105 kg and he is motionless in midair when struck near his center of mass?
    (c) Calculate the recoil velocity of the opponent’s 10.0-kg head if hit in this manner, assuming the head does not initially transfer significant momentum to the boxer’s body.
    (d) Discuss the implications of your answers for parts (b) and (c).
  5. Professional Application
    Suppose a child drives a bumper car head on into the side rail, which exerts a force of 4000 N on the car for 0.200 s.
    (a) What impulse is imparted by this force?
    (b) Find the final velocity of the bumper car if its initial velocity was 2.80 m/s and the car plus driver have a mass of 200 kg. You may neglect friction between the car and floor.
    Solution: (a) [latex]\text{800 kg}\cdot \text{m/s}[/latex] away from the wall; (b) [latex]1\text{.}\text{20 m/s}[/latex] away from the wall
  6. Professional Application
    One hazard of space travel is debris left by previous missions. There are several thousand objects orbiting Earth that are large enough to be detected by radar, but there are far greater numbers of very small objects, such as flakes of paint. Calculate the force exerted by a 0.100-mg chip of paint that strikes a spacecraft window at a relative speed of [latex]4\text{.}\text{00}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{m/s}[/latex], given the collision lasts [latex]6\text{.}\text{00}×{\text{10}}^{–8}\phantom{\rule{0.25em}{0ex}}s[/latex].
  7. Professional Application
    A 75.0-kg person is riding in a car moving at 20.0 m/s when the car runs into a bridge abutment.
    (a) Calculate the average force on the person if he is stopped by a padded dashboard that compresses an average of 1.00 cm.
    (b) Calculate the average force on the person if he is stopped by an air bag that compresses an average of 15.0 cm.
    Solution: (a) [latex]1\text{.}\text{50}×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}N[/latex] away from the dashboard; (b) [latex]1\text{.}\text{00}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}N[/latex] away from the dashboard
  8. Professional Application
    Military rifles have a mechanism for reducing the recoil forces of the gun on the person firing it. An internal part recoils over a relatively large distance and is stopped by damping mechanisms in the gun. The larger distance reduces the average force needed to stop the internal part.
    (a) Calculate the recoil velocity of a 1.00-kg plunger that directly interacts with a 0.0200-kg bullet fired at 600 m/s from the gun.
    (b) If this part is stopped over a distance of 20.0 cm, what average force is exerted upon it by the gun?
    (c) Compare this to the force exerted on the gun if the bullet is accelerated to its velocity in 10.0 ms (milliseconds).
  9. A cruise ship with a mass of [latex]1\text{.}\text{00}×{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}\text{kg}[/latex] strikes a pier at a speed of 0.750 m/s. It comes to rest 6.00 m later, damaging the ship, the pier, and the tugboat captain’s finances. Calculate the average force exerted on the pier using the concept of impulse. (Hint: First calculate the time it took to bring the ship to rest.)
    Solution: [latex]4\text{.}\text{69}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{N}[/latex] in the boat’s original direction of motion
  10. Calculate the final speed of a 110-kg rugby player who is initially running at 8.00 m/s but collides head-on with a padded goalpost and experiences a backward force of [latex]1\text{.}\text{76}×{\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{N}[/latex] for [latex]5\text{.}\text{50}×{\text{10}}^{\text{–2}}\phantom{\rule{0.25em}{0ex}}\text{s}[/latex].
  11. Water from a fire hose is directed horizontally against a wall at a rate of 50.0 kg/s and a speed of 42.0 m/s. Calculate the magnitude of the force exerted on the wall, assuming the water’s horizontal momentum is reduced to zero.
    Solution: [latex]2\text{.}\text{10}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}N[/latex] away from the wall
  12. A 0.450-kg hammer is moving horizontally at 7.00 m/s when it strikes a nail and comes to rest after driving the nail 1.00 cm into a board.
    (a) Calculate the duration of the impact.
    (b) What was the average force exerted on the nail?
  13. Starting with the definitions of momentum and kinetic energy, derive an equation for the kinetic energy of a particle expressed as a function of its momentum.
    Solution: [latex]\begin{array}{}\mathbf{p}=m\mathbf{v}⇒{p}^{2}={m}^{2}{v}^{2}⇒\frac{{p}^{2}}{m}={\mathrm{mv}}^{2}\\ ⇒\frac{{p}^{2}}{2m}=\frac{1}{2}{\mathrm{mv}}^{2}=\text{KE}\\ \text{KE}=\frac{{p}^{2}}{2m}\end{array}[/latex]
  14. A ball with an initial velocity of 10 m/s moves at an angle [latex]\text{60º}[/latex] above the \(+x[/latex]-direction. The ball hits a vertical wall and bounces off so that it is moving [latex]\text{60º}[/latex] above the [latex]-x[/latex]-direction with the same speed. What is the impulse delivered by the wall?
  15. When serving a tennis ball, a player hits the ball when its velocity is zero (at the highest point of a vertical toss). The racquet exerts a force of 540 N on the ball for 5.00 ms, giving it a final velocity of 45.0 m/s. Using these data, find the mass of the ball.
    Solution: 60.0 g
  16. A punter drops a ball from rest vertically 1 meter down onto his foot. The ball leaves the foot with a speed of 18 m/s at an angle [latex]\text{55º}[/latex] above the horizontal. What is the impulse delivered by the foot (magnitude and direction)?
  17. *Calculate the final speed of a 110-kg rugby player who is initially running at 8.00 m/s but collides head-on with a padded goalpost and experiences a backward force of 1.76×104N for 5.50×10−2s.
  18. *Water from a fire hose is directed horizontally against a wall at a rate of 50.0 kg/s and a speed of 42.0 m/s. Calculate the force exerted on the wall, assuming the water’s horizontal momentum is reduced to zero.
    Solution: 2.10×103N
  19. *A 0.450-kg hammer is moving horizontally at 7.00 m/s when it strikes a nail and comes to rest after driving the nail 1.00 cm into a board. Assume constant acceleration of the hammer-nail pair.
    (a) Calculate the duration of the impact.
    (b) What was the average force exerted on the nail?
  20. The x-component of a force on a 46-g golf ball by a 7-iron versus time is plotted in the following figure:A graph of F sub x in Newtons as a function of time in milliseconds. The horizontal axis ranges from 0 to 100 and the vertical axis rages from 0 to 30. The graph starts at 0 and rises to 30 N at time 50 millisecnds. It is then constant at 30 N until t = 100 when it drops to 0.(a) Find the x-component of the impulse during the intervals
    i. [0, 50 ms], and
    ii.     [50 ms, 100 ms]
    (b)     Find the change in the x-component of the momentum during the intervals
    i. [0, 50 ms], and
    ii. [50 ms, 100 ms]
  21. *A hockey puck of mass 150 g is sliding due east on a frictionless table with a speed of 10 m/s. Suddenly, a constant force of magnitude 5 N and direction due north is applied to the puck for 1.5 s. Find the north and east components of the momentum at the end of the 1.5-s interval.A puck is shown with force F equals 5.0 N north and v sub I = 10 meters per second east.Solution: px = 10 kg⋅m/s py= 20 kg⋅m/s

  22. *A ball of mass 250 g is thrown with an initial velocity of 25 m/s at an angle of 30° with the horizontal direction. Ignore air resistance. What is the momentum of the ball after 0.2 s? (Do this problem by finding the components of the momentum first, and then constructing the magnitude and direction of the momentum vector from the components.)

    A baseball has v sub I = 25 meters per second v hat at an angle of 30 degrees above the horizontal.

    6.3 Conservation of Momentum

    1. Professional Application
      Train cars are coupled together by being bumped into one another. Suppose two loaded train cars are moving toward one another, the first having a mass of 150,000 kg and a velocity of 0.300 m/s, and the second having a mass of 110,000 kg and a velocity of [latex]-0\text{.}\text{120 m/s}[/latex]. (The minus indicates direction of motion.) What is their final velocity?
      Solution: 0.122 m/s
    2. Suppose a clay model of a koala bear has a mass of 0.200 kg and slides on ice at a speed of 0.750 m/s. It runs into another clay model, which is initially motionless and has a mass of 0.350 kg. Both being soft clay, they naturally stick together. What is their final velocity?
    3. Professional Application
      Consider the following question: A car moving at 10 m/s crashes into a tree and stops in 0.26 s. Calculate the force the seatbelt exerts on a passenger in the car to bring him to a halt. The mass of the passenger is 70 kg. Would the answer to this question be different if the car with the 70-kg passenger had collided with a car that has a mass equal to and is traveling in the opposite direction and at the same speed? Explain your answer.
      Solution: In a collision with an identical car, momentum is conserved. Afterwards [latex]{v}_{\text{f}}=0[/latex] for both cars. The change in momentum will be the same as in the crash with the tree. However, the force on the body is not determined since the time is not known. A padded stop will reduce injurious force on body.
    4. What is the velocity of a 900-kg car initially moving at 30.0 m/s, just after it hits a 150-kg deer initially running at 12.0 m/s in the same direction? Assume the deer remains on the car.
    5. A 1.80-kg falcon catches a 0.650-kg dove from behind in midair. What is their velocity after impact if the falcon’s velocity is initially 28.0 m/s and the dove’s velocity is 7.00 m/s in the same direction?
      Solution: 22.4 m/s in the same direction as the original motion
    6. *Train cars are coupled together by being bumped into one another. Suppose two loaded train cars are moving toward one another, the first having a mass of 1.50×105kg and a velocity of (0.30m/s), and the second having a mass of 1.10×105kg and a velocity of −0.12m/s. What is their final velocity?Two train cars are shown moving toward each other. The car on the left is moving with v sub i 1 equals 0.30 meters per second i hat to the right, and the car on the right is moving with v sub i 2 equals -0.12 meters per second i hat to the left.
      Solution: 0.122m/s
    7. *Two identical pucks collide elastically on an air hockey table. Puck 1 was originally at rest; puck 2 has an incoming speed of 6.00 m/s and scatters at an angle of 30° with respect to its incoming direction. What is the velocity (magnitude and direction) of puck 1 after the collision?
      Two sets of red and blue hockey pucks are shown. The first row has a blue hockey puck with an arrow pointing left toward a red hockey puck. The second row shows a similar blue puck with a shorter arrow pointing left toward a red hockey puck. The red hockey puck also has an arrow pointing to its left.
    8. *The figure below shows a bullet of mass 200 g traveling horizontally towards the east with speed 400 m/s, which strikes a block of mass 1.5 kg that is initially at rest on a frictionless table.
      A drawing of a block on a table, and a bullet headed toward it.
      After striking the block, the bullet is embedded in the block and the block and the bullet move together as one unit.
      (a) What is the magnitude and direction of the velocity of the block/bullet combination immediately after the impact?
      (b) What is the magnitude and direction of the impulse by the block on the bullet?
      (c) What is the magnitude and direction of the impulse from the bullet on the block?
      (d) If it took 3 ms for the bullet to change the speed from 400 m/s to the final speed after impact, what is the average force between the block and the bullet during this time?
      Solution: (a) 47 m/s in the bullet to block direction; (b)70.6 N⋅ s, toward the bullet; (c) 70.6 N•s, toward the block; ( 

      d) magnitude is 2.35×104N

    9. *A 20-kg child is coasting at 3.3 m/s over flat ground in a 4.0-kg wagon. The child drops a 1.0-kg ball out the back of the wagon. What is the final speed of the child and wagon?
    10. *Two figure skaters are coasting in the same direction, with the leading skater moving at 5.5 m/s and the trailing skating moving at 6.2 m/s. When the trailing skater catches up with the leading skater, he picks her up without applying any horizontal forces on his skates. If the trailing skater is 50% heavier than the 50-kg leading skater, what is their speed after he picks her up?

    6.4 Elastic Collisions in One Dimension

    1. Two identical objects (such as billiard balls) have a one-dimensional collision in which one is initially motionless. After the collision, the moving object is stationary and the other moves with the same speed as the other originally had. Show that both momentum and kinetic energy are conserved.
    2. Professional Application
      Two piloted satellites approach one another at a relative speed of 0.250 m/s, intending to dock. The first has a mass of [latex]4\text{.}\text{00}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{kg}[/latex], and the second a mass of [latex]7\text{.}\text{50}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{kg}[/latex]. If the two satellites collide elastically rather than dock, what is their final relative velocity?
      Solution: 0.250 m/s
    3. A 70.0-kg ice hockey goalie, originally at rest, catches a 0.150-kg hockey puck slapped at him at a velocity of 35.0 m/s. Suppose the goalie and the ice puck have an elastic collision and the puck is reflected back in the direction from which it came. What would their final velocities be in this case?
    4. *A 5.50-kg bowling ball moving at 9.00 m/s collides with a 0.850-kg bowling pin, which is scattered at an angle to the initial direction of the bowling ball and with a speed of 15.0 m/s.
      (a) Calculate the final velocity (magnitude and direction) of the bowling ball.
      (b) Is the collision elastic?
      Solution: (a) 6.80 m/s, 5.33°; (b) yes (calculate the ratio of the initial and final kinetic energies)
    5. *A 90.0-kg ice hockey player hits a 0.150-kg puck, giving the puck a velocity of 45.0 m/s. If both are initially at rest and if the ice is frictionless, how far does the player recoil in the time it takes the puck to reach the goal 15.0 m away?
      Solution: 2.5 cm
    6. *You are standing on a very slippery icy surface and throw a 1-kg football horizontally at a speed of 6.7 m/s. What is your velocity when you release the football? Assume your mass is 65 kg.
    7. *A 35-kg child sleds down a hill and then coasts along the flat section at the bottom, where a second 35-kg child jumps on the sled as it passes by her. If the speed of the sled is 3.5 m/s before the second child jumps on, what is its speed after she jumps on?
      Solution: 1.8 m/s
    8. *A boy sleds down a hill and onto a frictionless ice-covered lake at 10.0 m/s. In the middle of the lake is a 1000-kg boulder. When the sled crashes into the boulder, he is propelled over the boulder and continues sliding over the ice. If the boy’s mass is 40.0 kg and the sled’s mass is 2.50 kg, what is the speed of the sled and the boulder after the collision?

    6.5 Inelastic Collisions in One Dimension

    1. A 0.240-kg billiard ball that is moving at 3.00 m/s strikes the bumper of a pool table and bounces straight back at 2.40 m/s (80% of its original speed). The collision lasts 0.0150 s.
      (a) Calculate the average force exerted on the ball by the bumper.
      (b) How much kinetic energy in joules is lost during the collision? (c) What percent of the original energy is left?
      Solution: (a) 86.4 N perpendicularly away from the bumper; (b) 0.389 J; (c) 64.0%
    2. During an ice show, a 60.0-kg skater leaps into the air and is caught by an initially stationary 75.0-kg skater.
      (a) What is their final velocity assuming negligible friction and that the 60.0-kg skater’s original horizontal velocity is 4.00 m/s?
      (b) How much kinetic energy is lost?
    3. Professional Application
      Using the data from a 110-kg football player running at 8.00 m/s catching a 0.410-kg football that has a speed of 25.0 m/s, and assuming that the football player catches the ball with his feet off the ground with both of them moving horizontally, calculate:
      (a) the final velocity if the ball and player are going in the same direction and
      (b) the loss of kinetic energy in this case.
      (c) Repeat parts (a) and (b) for the situation in which the ball and the player are going in opposite directions. Might the loss of kinetic energy be related to how much it hurts to catch the pass?
      Solution: (a) 8.06 m/s; (b) -56.0 J; (c)(i) 7.88 m/s; (ii) -223 J
    4. A battleship that is [latex]6\text{.}\text{00}×{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}\text{kg}[/latex] and is originally at rest fires a 1100-kg artillery shell horizontally with a velocity of 575 m/s.
      (a) If the shell is fired straight aft (toward the rear of the ship), there will be negligible friction opposing the ship’s recoil. Calculate its recoil velocity.
      (b) Calculate the increase in internal kinetic energy (that is, for the ship and the shell). This energy is less than the energy released by the gun powder—significant heat transfer occurs.
    5. Professional Application
      A 30,000-kg freight car is coasting at 0.850 m/s with negligible friction under a hopper that dumps 110,000 kg of scrap metal into it.
      (a) What is the final velocity of the loaded freight car?
      (b) How much kinetic energy is lost?
    6. A 0.0250-kg bullet is accelerated from rest to a speed of 550 m/s in a 3.00-kg rifle. The pain of the rifle’s kick is much worse if you hold the gun loosely a few centimeters from your shoulder rather than holding it tightly against your shoulder.
      (a) Calculate the recoil velocity of the rifle if it is held loosely away from the shoulder.
      (b) How much kinetic energy does the rifle gain?
      (c) What is the recoil velocity if the rifle is held tightly against the shoulder, making the effective mass 28.0 kg?
      (d) How much kinetic energy is transferred to the rifle-shoulder combination? The pain is related to the amount of kinetic energy, which is significantly less in this latter situation.
      (e) Calculate the momentum of a 110-kg football player running at 8.00 m/s. Compare the player’s momentum with the momentum of a hard-thrown 0.410-kg football that has a speed of 25.0 m/s. Discuss its relationship to this problem.
      Solution: (a) 4.58 m/s away from the bullet; (b) 31.5 J; (c) –0.491 m/s; (d) 3.38 J
    7. Professional Application
      Two football players collide head-on in midair while trying to catch a thrown football. The first player is 95.0 kg and has an initial velocity of 6.00 m/s, while the second player is 115 kg and has an initial velocity of –3.50 m/s. What is their velocity just after impact if they cling together?
    8. What is the speed of a garbage truck that is [latex]1\text{.}\text{20}×{\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{kg}[/latex] and is initially moving at 25.0 m/s just after it hits and adheres to a trash can that is 80.0 kg and is initially at rest?
      Solution: 24.8 m/s
    9. During a circus act, an elderly performer thrills the crowd by catching a cannon ball shot at him. The cannon ball has a mass of 10.0 kg and the horizontal component of its velocity is 8.00 m/s when the 65.0-kg performer catches it. If the performer is on nearly frictionless roller skates, what is his recoil velocity?
    10. (a) During an ice skating performance, an initially motionless 80.0-kg clown throws a fake barbell away. The clown’s ice skates allow her to recoil frictionlessly. If the clown recoils with a velocity of 0.500 m/s and the barbell is thrown with a velocity of 10.0 m/s, what is the mass of the barbell?
      (b) How much kinetic energy is gained by this maneuver?
      (c) Where does the kinetic energy come from?
      Solution: (a) 4.00 kg; (b) 210 J; (c) The clown does work to throw the barbell, so the kinetic energy comes from the muscles of the clown. The muscles convert the chemical potential energy of ATP into kinetic energy.

    Additional Problems

    1. *Two 70-kg canoers paddle in a single, 50-kg canoe. Their paddling moves the canoe at 1.2 m/s with respect to the water, and the river they’re in flows at 4 m/s with respect to the land. What is their momentum with respect to the land?
    2. *Which has a larger magnitude of momentum: a 3000-kg elephant moving at 40 km/h or a 60-kg cheetah moving at 112 km/h?
      Solution: the elephant has a higher momentum
    3. *A driver applies the brakes and reduces the speed of her car by 20%, without changing the direction in which the car is moving. By how much does the car’s momentum change?
    4. *You friend claims that momentum is mass multiplied by velocity, so things with more mass have more momentum. Do you agree? Explain.
      Solution: Answers may vary. The first clause is true, but the second clause is not true in general because the velocity of an object with small mass may be large enough so that the momentum of the object is greater than that of a larger-mass object with a smaller velocity.
    5. *Dropping a glass on a cement floor is more likely to break the glass than if it is dropped from the same height on a grass lawn. Explain in terms of the impulse.
    6. *Your 1500-kg sports car accelerates from 0 to 30 m/s in 10 s. What average force is exerted on it during this acceleration?
      Solution: 4.5×103N
    7. *A ball of mass m is dropped. What is the formula for the impulse exerted on the ball from the instant it is dropped to an arbitrary time τ later? Ignore air resistance.
    8. *Two hockey players approach each other head on, each traveling at the same speed vi. They collide and get tangled together, falling down and moving off at a speed vi/5. What is the ratio of their masses?
    9. *You are coasting on your 10-kg bicycle at 15 m/s and a 5.0-g bug splatters on your helmet. The bug was initially moving at 2.0 m/s in the same direction as you. If your mass is 60 kg,
      (a) what is the initial momentum of you plus your bicycle?
      (b) What is the initial momentum of the bug?
      (c) What is your change in velocity due to the collision with the bug?
      (d) What would the change in velocity have been if the bug were traveling in the opposite direction?
      Solution: (a) 1.1×103kg⋅m/s; (b) 0.010kg⋅m/s; (c) −0.00093m/s; (d) −0.0012m/s
    10. *A 100-kg astronaut finds himself separated from his spaceship by 10 m and moving away from the spaceship at 0.1 m/s. To get back to the spaceship, he throws a 10-kg tool bag away from the spaceship at 5.0 m/s. How long will he take to return to the spaceship?
    11. *A child sleds down a hill and collides at 5.6 m/s into a stationary sled that is identical to his. The child is launched forward at the same speed, leaving behind the two sleds that lock together and slide forward more slowly. What is the speed of the two sleds after this collision?
      Solution: 2.8 m/s
    12. *For the preceding problem, find the final speed of each sled for the case of an elastic collision.
    13. *A 90-kg football player jumps vertically into the air to catch a 0.50-kg football that is thrown essentially horizontally at him at 17 m/s. What is his horizontal speed after catching the ball?
      Solution: 0.094 m/s
    14. *Three skydivers are plummeting earthward. They are initially holding onto each other, but then push apart. Two skydivers of mass 70 and 80 kg gain horizontal velocities of 1.2 m/s north and 1.4 m/s southeast, respectively. What is the horizontal velocity of the third skydiver, whose mass is 55 kg?

    Challenge Problems

    1. A 65-kg person jumps from the first floor window of a burning building and lands almost vertically on the ground with a horizontal velocity of 3 m/s and vertical velocity of -9m/s. Upon impact with the ground he is brought to rest in a short time. The force experienced by his feet depends on whether he keeps his knees stiff or bends them. Find the force on his feet in each case.
      A drawing of a person near the ground. His velocity vector is directed down and slightly to the left and is given as 3.0 meters per second i hat minus 9.0 meters per second j hat. The x y directions are shown for reference, with x to the right and y upward.(a) First find the impulse on the person from the impact on the ground. Calculate both its magnitude and direction.
      (b)       Find the average force on the feet if the person keeps his leg stiff and straight and his center of mass drops by only 1 cm vertically and 1 cm horizontally during the impact.
      (c)       Find the average force on the feet if the person bends his legs throughout the impact so that his center of mass drops by 50 cm vertically and 5 cm horizontally during the impact.
      (d)       Compare the results of part (b) and (c), and draw conclusions about which way is better.
      You will need to find the time the impact lasts by making reasonable assumptions about the deceleration. Although the force is not constant during the impact, working with constant average force for this problem is acceptable.
      Solution: (a) 617N⋅s, 108°; (b) Fx=2.91×104N , Fy=2.6×105 N; (c)       

       Fx=5850N, Fy=5265N

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      Introduction to Biomechanics Copyright © 2022 by Rob Pryce is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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