Worksheet: Collisions and Conservation of Momentum

In this worksheet, we will practice applying the law of conservation of momentum to study collisions in one dimension and differentiating between elastic and inelastic collisions.

Q1:

A ball of mass 3 kg, moving in a straight line at 32 cm/s, collided with another ball of mass 2.5 kg that was at rest. Given that the two balls coalesced into one body, determine the speed of this new body.

  • A32 cm/s
  • B17.45 cm/s
  • C14.55 cm/s
  • D38.4 cm/s

Q2:

Two smooth spheres of masses 83 g and 37 g were moving in a straight line. At time 𝑑 seconds, where 𝑑β‰₯0, the spheres’ displacements relative to a fixed point are given by sc=[(165𝑑)]cm and sc=[(βˆ’195𝑑)]cm, respectively, where c is a fixed unit vector. Given that the two spheres collided and coalesced into one body, determine the speed 𝑣 of this composite body and the magnitude of the impulse 𝐼 between the spheres.

  • A 𝑣 = 8 4 / c m s , 𝐼 = 2 0 , 6 6 7 β‹… / g c m s
  • B 𝑣 = 5 4 / c m s , 𝐼 = 1 8 , 1 7 7 β‹… / g c m s
  • C 𝑣 = 5 4 / c m s , 𝐼 = 6 , 7 2 3 β‹… / g c m s
  • D 𝑣 = 8 4 / c m s , 𝐼 = 6 , 7 2 3 β‹… / g c m s
  • E 𝑣 = 5 4 / c m s , 𝐼 = 9 , 2 1 3 β‹… / g c m s

Q3:

A builder is hammering nails into a wall. The hammer has a mass of 3.3 kg, and each nail has a mass of 308 g. Given that the hammer hits each stationary nail at a speed of 8.2 m/s, use the principle of conservation of momentum to find the speed of the hammer and nail directly after the impact.

Q4:

Two spheres, 𝐴 and 𝐡, are moving in a straight line on a smooth horizontal plane in opposite directions at 7.17 m/s. If their masses are 4π‘š and 8π‘š, respectively, find the velocity π‘£οŒ οŒ‘ of sphere 𝐴 relative to sphere 𝐡. Given that the two bodies coalesce on impact into one body, find the speed 𝑣 of this new body just after the collision.

  • A 𝑣 = 0   , 𝑣 = 7 . 1 7 / m s
  • B 𝑣 = 1 4 . 3 4 /   m s , 𝑣 = 2 . 3 9 / m s
  • C 𝑣 = 1 4 . 3 4 /   m s , 𝑣 = 7 . 1 7 / m s
  • D 𝑣 = 0   , 𝑣 = 2 . 3 9 / m s

Q5:

A ball of mass 60 g started accelerating from rest at 7 cm/s2. At the same moment, another ball of mass 40 g, which was 450 cm ahead of the first ball, started moving in the same direction as the first ball at a constant speed of 90 cm/s. Then the two bodies collided and coalesced into one body. Find the speed of this body directly after the collision.

Q6:

Two smooth spheres, 𝐴 and 𝐡, of masses 160 g and 40 g, respectively, are moving in opposite directions along a horizontal straight line. Sphere 𝐴 was moving at a constant speed of 95 cm/s, and sphere 𝐡 was initially moving at 75 cm/s while accelerating at 5 cm/s2. After sphere 𝐡 covered a distance of 340 cm, the two spheres collided and coalesced into one body. Determine the speed of this body right after the collision.

Q7:

A sphere of mass 28 g was moving horizontally in a straight line at 319 cm/s when it collided with another sphere of mass 30 g which had been at rest. If the spheres coalesced into one body as a result of the impact, and the body continued moving under the influence of a constant resistive force of 29 g-wt, determine the distance it covered after the collision until it came to rest. Consider the acceleration due to gravity to be 𝑔=9.8/ms.

Q8:

A body of mass 0.6 kg, moving in a straight line at 33 cm/s, crashed into another body of mass 0.5 kg that was at rest. If they coalesced into one body, and this new body moved 20 cm before it came to rest, find the magnitude of the force resisting this body’s movement assuming that it was constant.

Q9:

A ball of mass 125 g moving at a constant speed of 50 m/s passed a certain point and, 3 minutes later, another ball of mass 75 g passed the same point. It was moving at 80 cm/s and accelerating at 4 cm/s2. The two balls collided and coalesced into one ball. Determine the time 𝑑 taken for the second ball to hit the first and the speed of the coalesced body after the impact.

  • A 𝑑 = 4 3 . 8 3 2 s , 𝑣 = 1 2 6 . 9 9 8 / c m s
  • B 𝑑 = 6 0 s , 𝑣 = 2 1 8 . 7 5 / c m s
  • C 𝑑 = 6 0 s , 𝑣 = 1 5 1 . 2 5 / c m s
  • D 𝑑 = 4 2 . 0 4 s , 𝑣 = 1 7 3 . 8 5 1 / c m s
  • E 𝑑 = 4 2 . 0 4 s , 𝑣 = 1 2 4 . 3 1 / c m s

Q10:

Two spheres are moving along a straight line. One has a mass of π‘š and is moving at speed 𝑣, whereas the other one has a mass of 10 g and is moving at 36 cm/s. If the two spheres were moving in the same direction when they collided, they would coalesce into one body and move at 30 cm/s in the same direction. However, if they were moving in opposite directions, they would coalesce into one body which would move at 6 cm/s in the direction the first sphere had been traveling. Find π‘š and 𝑣.

  • A π‘š = 5 g , 𝑣 = 1 0 8 / c m s
  • B π‘š = 4 5 g , 𝑣 = 1 2 / c m s
  • C π‘š = 2 0 g , 𝑣 = 2 7 / c m s
  • D π‘š = 3 0 g , 𝑣 = 1 8 / c m s

Q11:

On a smooth plane inclined at 30∘ to the horizontal, 𝐴𝐡 is a line segment parallel to a line of greatest slope of the plane. The point 𝐴 is at the top of the plane, and the distance between 𝐴 and 𝐡 is 7 m. A sphere of mass 45 g was left to roll down the plane from the point 𝐴. At the same moment, another sphere of mass 165 g was projected up the line segment from point 𝐡 at 7 m/s. Given that, when the two spheres collided, they coalesced into one body, find the speed 𝑣 of this combined body and the maximum distance 𝑙 that the body will move up the plane after the collision stating the answer to the nearest two decimal places. Consider the acceleration due to gravity to be 𝑔=9.8/ms.

  • A 𝑣 = 2 . 8 / m s , 𝑙 = 4 0 . 0 0 c m
  • B 𝑣 = 3 . 2 5 / m s , 𝑙 = 1 0 7 . 7 8 c m
  • C 𝑣 = 0 . 3 / m s , 𝑙 = 0 . 9 2 c m
  • D 𝑣 = 0 . 6 / m s , 𝑙 = 3 . 6 7 c m

Q12:

A railway carriage of mass 11 metric tons started moving down a plane from rest. The plane was inclined to the horizontal at an angle whose sine is 150, and the resistance to the carriage’s motion was 16 kg-wt per tonne of the carriage’s mass. The carriage reached the bottom of the plane after covering a distance of 144 m, and then it collided with another carriage of the same mass that was at rest. Given that the two carriages coupled together as a result of the impact then came to rest 50 seconds later, determine the distance covered by the two carriages on the horizontal road. Consider the acceleration due to gravity to be 𝑔=9.8/ms.

Q13:

A body of mass 5 kg was moving at 18 cm/s when it collided with another body of mass 1 kg moving at 27 cm/s in the opposite direction. The two bodies coalesced into one body. Then, this body collided with a third body of mass 12 kg which was at rest. As a result, this body coalesced into the other, forming one body from all three. Find the speed of this last compound body after the collision.

Q14:

Three smooth spheres 𝐴, 𝐡, and 𝐢 of masses 430, 140, and 840 grams, respectively, were placed on a smooth horizontal table along a straight line such that 𝐡 lies between 𝐴 and 𝐢. Sphere 𝐴 was projected at 57 cm/s toward 𝐡. When they collided, they coalesced into one body that kept moving toward 𝐢. When this new body collided with 𝐢, it rebounded at 6 cm/s. Find the speed of sphere 𝐢 after the impact.

Q15:

A sphere of mass 68 g was projected along a rough horizontal plane at 23 m/s. The resistance to the motion of the sphere was 114 of the sphere’s weight. Then, 10 seconds later, it collided with another sphere of the same mass moving at 18 m/s in the opposite direction. As a result of the collision, the two spheres coalesced into one body. Calculate the speed of this new body immediately after the impact. Take 𝑔=9.8/ms.

Q16:

Two projectiles of the same mass of 50 kg were fired at 44 m/s toward a target of mass 400 kg that was moving away from the projectiles at 11 m/s. Both projectiles hit the target and sunk into it. This combined body was then hit again from the opposite direction by another projectile of mass 220 kg. Given that this body also coalesced with the target and, as a result of the impact, this new body came to rest, determine the speed of the last projectile which hit the body.

Q17:

A mechanical hammer of mass 632 kg fell from a height of 2.5 m onto a body of mass 474 kg. As a result, the body penetrated 20 cm into the ground. Immediately after the impact, the two bodies moved together with the same speed 𝑣. Given that the acceleration due to gravity is 9.8 m/s2, find 𝑣, and determine the resistance 𝑅 of the ground.

  • A 𝑣 = 4 / m s , 𝑅 = 1 1 , 2 8 1 . 2 N
  • B 𝑣 = 4 . 9 / m s , 𝑅 = 1 1 , 5 0 2 . 7 N
  • C 𝑣 = 4 / m s , 𝑅 = 5 5 , 0 7 8 . 8 N
  • D 𝑣 = 4 . 9 / m s , 𝑅 = 7 7 , 2 2 6 . 5 N

Q18:

A body of mass 1.8 kg was projected vertically upward from the ground at 14.7 m/s and 1 second later another body of mass 2.7 kg was projected vertically upward from the same point at 18.9 m/s. The two bodies collided and they coalesced into one body. Find the maximum height this compound body reached above the ground. Consider the acceleration due to gravity to be 𝑔=9.8/ms.

Q19:

A body 𝐴 of mass 50 g was falling vertically at 120 cm/s when it collided with a body 𝐡 of mass 40 g moving vertically upward at 700 cm/s. Body 𝐡 rebounded vertically downward at 140 cm/s, whereas body 𝐴 rebounded vertically upward. Then, 17 of a second later, body 𝐴 collided with another body, 𝐢, of mass 300 g moving vertically downward at 15 cm/s. The two bodies coalesced into one body, and the body continued moving. Find the speed of this compound body after the second collision. Consider the acceleration due to gravity to be 9.8 m/s2.

Q20:

Two bodies of masses 861 g and 287 g were moving toward each other along the same straight line at 8 m/s. When the two bodies collided, they coalesced into one body. Determine the speed of this new body.

Q21:

A bullet of mass 24 g was fired at 462 m/s toward a target of mass 1 kg which was at rest. After the impact, the target and the bullet moved together as one body. Given that it came to rest after covering a distance of 105 cm, determine the resistance to the body’s motion, assuming that it was constant.

  • A 5 . 3 6 Γ— 1 0  dynes
  • B 2 . 2 3 3 Γ— 1 0  dynes
  • C 1 . 3 4 Γ— 1 0  dynes
  • D 5 . 7 1 7 Γ— 1 0  dynes

Q22:

Two spheres were projected one after the other along the same straight line and in the same direction. The mass of the first sphere was 230 g, and its speed was 14 cm/s, while the mass of the other was 345 g, and its speed was 25 cm/s. Given that the spheres coalesced into one body when they collided, determine the speed of this compound body.

Q23:

A body started falling from a point that is 104.4 m above the ground. At the same time, another body was projected vertically upward at 40.6 m/s from the ground. The two bodies met at a point 𝑑 meters above the ground at time 𝑑. Find 𝑑 and 𝑑, and determine whether the two bodies met while moving in the same or opposite directions. Take 𝑔=9.8/ms.

  • A 𝑑 = 1 8 7 s , 𝑑 = 7 2 m , same direction
  • B 𝑑 = 1 8 7 s , 𝑑 = 3 2 . 4 m , opposite directions
  • C 𝑑 = 1 8 7 s , 𝑑 = 3 2 . 4 m , same direction
  • D 𝑑 = 1 8 7 s , 𝑑 = 7 2 m , opposite directions
  • E 𝑑 = 4 0 7 s , 𝑑 = 7 2 m , opposite directions

Q24:

A bullet was fired horizontally at 500 m/s toward a piece of wood. It hit the piece of wood and penetrated 20 cm deep before it stopped. If a similar bullet was fired at a similar target made of the same kind of wood but of thickness 11 cm, determine the speed at which it would exit the back of the target rounded to two decimal places.

Q25:

A bullet was fired horizontally at 900 m/s towards a thick piece of wood. The bullet struck the piece of wood and penetrated 9 cm deep before it stopped. If a similar bullet was fired at a similar target made of the same kind of wood but of thickness 6 cm, determine the minimum speed at which it needs to be fired to pass all the way through the target stating your answer to two decimal places.

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