A baseball has a mass of about 0.14 kg, and it is pitched towards home plate at a speed of about 46 m/s. If the bat exerts an average force of 9500 N for 1.7 ms, what is the final speed of the ball in m/s?

To find the final speed of the ball, we can use Newton's second law of motion, which states that force is equal to the rate of change of momentum. In this equation, momentum is defined as the mass of an object multiplied by its velocity.

The formula for momentum is:

Momentum (p) = mass (m) × velocity (v)

Given that the mass of the baseball is 0.14 kg and the initial velocity is 46 m/s, we can calculate the initial momentum (p_initial).

p_initial = m × v_initial
= 0.14 kg × 46 m/s
= 6.44 kg·m/s

Now, let's calculate the force exerted by the bat. The average force (F) is given as 9500 N, and the time (t) is given as 1.7 ms. However, we need to convert the time from milliseconds to seconds.

t = 1.7 ms = 1.7 × 10^-3 s

To calculate the change in momentum (Δp) caused by the bat's force, we can use the formula:

Force (F) = Δp / Δt

Rearranging the formula, we get:

Δp = F × Δt

Now, let's substitute the values:

Δp = 9500 N × 1.7 × 10^-3 s
= 16.15 kg·m/s

The final momentum (p_final) of the ball is given by the initial momentum (p_initial) plus the change in momentum (Δp).

p_final = p_initial + Δp
= 6.44 kg·m/s + 16.15 kg·m/s
= 22.59 kg·m/s

Finally, to find the final velocity (v_final) of the ball, we divide the final momentum by the mass of the ball:

v_final = p_final / m
= 22.59 kg·m/s / 0.14 kg
≈ 161.36 m/s

Therefore, the final speed of the ball is approximately 161.36 m/s.