Question

Billiard ball A, with a mass mA​=0.117 kg and speed vA​=2.80 m/s, strikes ball B, initially at rest, with a mass mB​=0.135 kg. After the collision, ball A is deflected off at an angle of θA′​=30.0∘ with a speed vA′​=2.10 m/s, and ball B moves with a speed vB′​ at an angle θB′​ to the original direction of motion of ball A.

Taking the x-axis as the original direction of motion of ball A, choose the correct equation expressing the conservation of momentum for the components in the x-direction.

A. 0=mA​vA′​sinθA′​−mB​vB′​sinθB′​

B. mA​vA​=mA​vA′​cosθA′​+mB​vB′​cosθB′​

C. mA​vA​=mA​vA′​cosθA′​−mB​vB′​sinθB′​

D. 0=(mA​vA​+mB​vB′​)sinθB′​

Answer 1

The equation expressing the conservation of momentum for the components in the x-direction is mAvA = mAvA'cosθA' + mBvB'cosθB' which states that the initial momentum of ball A equals the sum of the momentum components of both balls after the collision along the x-axis. Hence, option (B) is correct.

Step-by-step explanation:

The correct equation expressing the conservation of momentum for the components in the x-direction when billiard ball A, with mass mA and speed vA, strikes ball B, which is initially at rest with mass mB, is given by option B. Here is the step-by-step explanation of how to find the correct equation:

• Conservation of momentum states that the total momentum before the collision equals the total momentum after the collision.
• In the x-direction, which is the original direction of motion for ball A, the momentum would be the mass times the velocity component in that direction.
• To find the x-direction velocity components after the collision, we use the cosine of the angle made with respect to the x-axis because the cosine function relates the adjacent side (the x-component in this case) to the hypotenuse in a right triangle.

Therefore, the correct expression of momentum conservation in the x-direction is:

mA​vA​ = mA​vA'​cosθA' + mB​vB'​cosθB'

This equation is equivalent to saying that the x-component of the velocity of ball A after the collision, plus the x-component of the velocity of ball B after the collision, equals the initial x-component of the velocity of ball A since ball B starts from rest.