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29 votes
29 votes
A 7.50 kg bowling ball moving

6.42 m/s strikes a 1.60 kg bowling
pin at rest. After, the ball moves
5.43 m/s at a 12.0° angle. What is
the x-component and y-component of the pin's final
velocity?

User Igouy
by
2.4k points

1 Answer

17 votes
17 votes

Answer:

Step-by-step explanation:

To solve this problem we need to know the direction in which the ball was moving to start with.

The answer will be different depending n the original angle of the ball's movement.

It might be reasonable to assume that the ball is meant to approach along the x-axis,

but if so, the initial speed of 6.42m/s would be irrelevant to the answer.

So I will solve the problem for the general case of two objects colliding at arbitrary angles, and

tell you how to specialize it for any assumption about the initial conditions.

Let

m1 = 7.5 kg be the mass of the ball,

m2 = 1.6 kg be the mass of the pin,

v1 = 6.42 m/s be the velocity of the ball before the strike,

v2 = 0 m/s be the velocity of the pin before the strike,

α1 be the angle of v1,

α2 be the angle of v2,

w1 be the velocity of the ball after the strike,

w2 = 14.8 m/s be the velocity of the pin after the strike,

β1 be the angle of w1,

β2 = -47° be the angle of w2.

By conservation of momentum:

m1v1 + m2v2 = m1w1 + m2w2

Since the velocities are vectors, the addition is vector addition, and the equality is vector equality.

"Vector equality" means that the x-coordinates are equal and the y-coordinates are equal.

The problem cares only about y-coordinates, specifically the y-coordinate of w1, which is w1sin(β1).

(In general, the y-coordinate of any vector is obtained by multiplying the vector's norm by the sine of its angle.)

Conservation of momentum in the y-coordinate is then

m1v1sin(α1) + m2v2sin(α2) = m1w1sin(β1) + m2w2sin(β2)

Expressing the sought quantity

w1sin(β1) = (m1v1sin(α1) + m2v2sin(α2) - m2w2sin(β2))/m1

Substituting known quantities:

w1sin(β1) = (7.5×6.42×sin(α1) + 1.6×0×sin(α2) - 1.6×14.8×sin(-47°))/7.5

= (48.15×sin(α1) + 17.3)/7.5

In the above expression we do not know α1.

If we assume that the ball is approaching along the x-axis then α1 = 0, and

w1sin(β1) = 17.3/7.5 = 2.3

Under that assumption the y-component of the ball's final velocity is 2.3 m/s;

being positive, it is opposite the direction of the pin.

User J Starr
by
3.3k points