Bailey is on the tenth frame of her recent bowling competition and she needs to pick up the last pin for a spare and the first place trophy. She rolls the 7.05-kg ball down the lane and it hits the 1.53-kg pin head on. the ball was moving at 8.24 m/s before the collision. The pin went flying forward at 13.2 m/s. Determine the post collision speed of the ball

momentum is conserved

Mb * V1b = (Mb * V2b) + (Mp * Vp)

initial momentum

= 7.05*8.24 + 0

final momentum
= 7.05 * v + 1.53*13.2

final momentum = initial momentum

To determine the post-collision speed of the ball, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision should be equal to the total momentum after the collision.

The momentum before the collision is given by the product of mass and velocity:
Momentum of the ball before collision = mass of the ball * velocity of the ball
Momentum of the ball before collision = 7.05 kg * 8.24 m/s
Momentum of the ball before collision = 58.182 kg·m/s

Similarly, the momentum of the pin before the collision is given by:
Momentum of the pin before collision = mass of the pin * velocity of the pin
Momentum of the pin before collision = 1.53 kg * 0 m/s (as it is initially at rest)
Momentum of the pin before collision = 0 kg·m/s

Since momentum is conserved, the total momentum after the collision is equal to the momentum before the collision:
Total momentum after collision = Momentum of the ball after collision + Momentum of the pin after collision

Let's assume the post-collision speed of the ball is Vf (final velocity).
So, Momentum of the ball after collision = mass of the ball * Vf
Momentum of the ball after collision = 7.05 kg * Vf

And, Momentum of the pin after collision = mass of the pin * velocity of the pin
Momentum of the pin after collision = 1.53 kg * 13.2 m/s

Therefore, according to the conservation of momentum:
Momentum of the ball after collision + Momentum of the pin after collision = Momentum of the ball before collision + Momentum of the pin before collision

7.05 kg * Vf + 1.53 kg * 13.2 m/s = 58.182 kg·m/s + 0 kg·m/s

Now, let's solve the equation to find the post-collision speed of the ball:
7.05 kg * Vf = 58.182 kg·m/s + (1.53 kg * 13.2 m/s)
7.05 kg * Vf = 58.182 kg·m/s + 20.196 kg·m/s
7.05 kg * Vf = 78.378 kg·m/s

Dividing both sides by 7.05 kg:
Vf = 78.378 kg·m/s / 7.05 kg
Vf ≈ 11.1 m/s

Therefore, the post-collision speed of the ball is approximately 11.1 m/s.

To determine the post-collision speed of the ball, we can use the principles of conservation of momentum.

The formula for conservation of momentum is:

(mass of object 1 * initial velocity of object 1) + (mass of object 2 * initial velocity of object 2) = (mass of object 1 * final velocity of object 1) + (mass of object 2 * final velocity of object 2)

In this scenario, the first object is the ball and the second object is the pin. Let's assign variables to the given values:

Mass of the ball (m1) = 7.05 kg
Initial velocity of the ball (v1i) = 8.24 m/s
Mass of the pin (m2) = 1.53 kg
Initial velocity of the pin (v2i) = 0 m/s (since it was at rest initially)

Final velocity of the ball (v1f) = Unknown
Final velocity of the pin (v2f) = 13.2 m/s (given)

Using the conservation of momentum formula, we can rewrite it with the assigned variables:

(m1 * v1i) + (m2 * v2i) = (m1 * v1f) + (m2 * v2f)

Substituting the values, we have:

(7.05 kg * 8.24 m/s) + (1.53 kg * 0 m/s) = (7.05 kg * v1f) + (1.53 kg * 13.2 m/s)

Calculating the equation:

(58.182 kg⋅m/s) + 0 = (7.05 kg * v1f) + (20.196 kg⋅m/s)

Now, let's isolate the final velocity of the ball (v1f):

(58.182 kg⋅m/s) - (20.196 kg⋅m/s) = 7.05 kg * v1f

37.986 kg⋅m/s = 7.05 kg * v1f

Dividing both sides of the equation by 7.05 kg:

v1f = (37.986 kg⋅m/s) / (7.05 kg)

v1f ≈ 5.377 m/s

Therefore, the post-collision speed of the ball is approximately 5.377 m/s.