A 7.50 kg bowling ball moving 6.42 m/s strikes a 1.60 kg bowling pin at rest. After, the pin moves 14.8 m/s at a -47.0° angle. What is the y-component of the ball's final velocity?

Answer 1

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.