Question

A 1,786 kg car stopped at a traffic light is struck from the rear by an 893-kg car. The two cars become entangled, moving along the same path as that of the originally moving car. If the smaller car were moving at 30.7 m/s before the collision, what is the velocity of the entangled cars after the collision? (Assume the smaller car initially moves in the positive direction.) SOLUTION Conceptualize This kind of collision is easily visualized, and one can predict that after the collision both cars will be moving in the same direction as that of the initially moving car. Because the initially moving car has only half the mass of the stationary car, we expect the final velocity of the cars to be relatively small. Categorize We identify the two cars as an isolated system in terms of momentum in the horizontal direction and apply the impulse approximation during the short time interval of the collision. The phrase "become entangled" tells us to categorize the collision as perfectly inelastic. Analyze The magnitude of the total momentum of the system before the collision is equal to that of the smaller car because the larger car is initially at rest. (Use the following as necessary: m?, m?, and v_f) Use the isolated system model for momentum: ??p???= 0 ? p_i = p_f ? m?v_i = (???????????)v_f Solve for v_f (in m/s) and substitute numerical values (Indicate the direction with the sign of your answer.): v_f = m?v_i / (m? + m?) = ??????????? m/s Finalize Because the final velocity is positive, the direction of the final velocity of the combination is the velocity of the initially moving car as predicted. The speed of the combination is also much lower than the initial speed of the moving car. EXERCISE (a) What is the loss of kinetic energy (K_i ? K_f, in kJ) in the situation described in the Example? (b) What if the 893 kg car actually moves backwards with a speed of 2.0 m/s right after the collision instead of having a perfectly inelastic collision? What is the velocity of the heavier car (in m/s) immediately after the collision? Use the same convention for positive direction as defined in the Example. (Indicate the direction with the sign of your answer.) (c) What is the loss of kinetic energy (in kJ) in this case?

A 1,786 kg car stopped at a traffic light is struck from the rear by an 893-kg car. The two cars become entangled, moving along the same path as that of the originally moving car. If the smaller car were moving at 30.7 m/s before the collision, what is the velocity of the entangled cars after the collision? (Assume the smaller car initially moves in the positive direction.)

SOLUTION

Conceptualize This kind of collision is easily visualized, and one can predict that after the collision both cars will be moving in the same direction as that of the initially moving car. Because the initially moving car has only half the mass of the stationary car, we expect the final velocity of the cars to be relatively small.

Categorize We identify the two cars as an isolated system in terms of momentum in the horizontal direction and apply the impulse approximation during the short time interval of the collision. The phrase "become entangled" tells us to categorize the collision as perfectly inelastic.

Analyze The magnitude of the total momentum of the system before the collision is equal to that of the smaller car because the larger car is initially at rest. (Use the following as necessary: m?, m?, and v_f)

Use the isolated system model for momentum:
??p???= 0 ? p_i = p_f ? m?v_i = (???????????)v_f

Solve for v_f (in m/s) and substitute numerical values (Indicate the direction with the sign of your answer.):
v_f = m?v_i / (m? + m?) = ??????????? m/s

Finalize Because the final velocity is positive, the direction of the final velocity of the combination is the velocity of the initially moving car as predicted. The speed of the combination is also much lower than the initial speed of the moving car.

EXERCISE

(a) What is the loss of kinetic energy (K_i ? K_f, in kJ) in the situation described in the Example?

(b) What if the 893 kg car actually moves backwards with a speed of 2.0 m/s right after the collision instead of having a perfectly inelastic collision? What is the velocity of the heavier car (in m/s) immediately after the collision? Use the same convention for positive direction as defined in the Example. (Indicate the direction with the sign of your answer.)

A 1,786 kg car stopped at a traffic light is struck from the rear by an 893-kg car. The two cars become entangled, moving along the same path as that of the originally moving car. If the smaller car were moving at 30.7 m/s before the collision, what is the velocity of the entangled cars after the collision? (Assume the smaller car initially moves in the positive direction.)

SOLUTION

Conceptualize This kind of collision is easily visualized, and one can predict that after the collision both cars will be moving in the same direction as that of the initially moving car. Because the initially moving car has only half the mass of the stationary car, we expect the final velocity of the cars to be relatively small.

Categorize We identify the two cars as an isolated system in terms of momentum in the horizontal direction and apply the impulse approximation during the short time interval of the collision. The phrase "become entangled" tells us to categorize the collision as perfectly inelastic.

Analyze The magnitude of the total momentum of the system before the collision is equal to that of the smaller car because the larger car is initially at rest. (Use the following as necessary: m?, m?, and vf)

Use the isolated system model for momentum:
??p???= 0 ? pi = pf ? m?vi = (???????????)vf

Solve for vf (in m/s) and substitute numerical values (Indicate the direction with the sign of your answer.):
vf = m?vi / (m? + m?) = ??????????? m/s

Finalize Because the final velocity is positive, the direction of the final velocity of the combination is the velocity of the initially moving car as predicted. The speed of the combination is also much lower than the initial speed of the moving car.

EXERCISE

(a) What is the loss of kinetic energy (Ki ? Kf, in kJ) in the situation described in the Example?

(b) What if the 893 kg car actually moves backwards with a speed of 2.0 m/s right after the collision instead of having a perfectly inelastic collision? What is the velocity of the heavier car (in m/s) immediately after the collision? Use the same convention for positive direction as defined in the Example. (Indicate the direction with the sign of your answer.)

(c) What is the loss of kinetic energy (in kJ) in this case?

Added by Catherine R.

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