A 0.145-kg baseball pitched horizontally at 27 m/s strikes a bat and is popped straight up to a height of 38 m. If the contact time between the bat and the ball is 2.35 ms, calculate the average force [exerted by the bat on the ball] during contact. [Let the positive axis lie along the line from the batter to the pitcher, with the batter at the origin.]
To calculate the average force exerted by the bat on the ball during contact, you can use the impulse-momentum principle. The impulse experienced by an object is equal to the change in momentum it undergoes.
Here's how you can solve the problem step by step:
1. Calculate the initial momentum of the baseball. Momentum (p) is given by the product of mass (m) and velocity (v).
- Mass (m) = 0.145 kg
- Velocity (v) = 27 m/s
- Initial momentum (p) = m * v
2. The baseball is popped straight up, reaching a height of 38 m. This implies that the final velocity in the vertical direction (vf_y) is zero (since it momentarily stops at the highest point).
- Final velocity in the vertical direction (vf_y) = 0 m/s
3. Calculate the change in momentum in the vertical direction (Δp_y). Since the baseball is moving horizontally, there is no change in momentum in that direction.
- Δp_y = 0 - m * vf_y
4. Use the contact time (Δt) and the change in momentum (Δp_y) to calculate the average force (F_avg) during contact using the impulse-momentum principle.
- F_avg = Δp_y / Δt
5. Substitute the values into the equation and calculate the average force.
- F_avg = Δp_y / Δt = (0 - m * vf_y) / Δt
Now, let's plug in the values to calculate the average force:
- Mass (m) = 0.145 kg
- Final velocity in the vertical direction (vf_y) = 0 m/s
- Contact time (Δt) = 2.35 ms = 0.00235 s
F_avg = (0 - 0.145 kg * 0 m/s) / 0.00235 s
Therefore, the average force exerted by the bat on the ball during contact is zero since the change in momentum in the vertical direction is zero.
To calculate the average force exerted by the bat on the ball during contact, we can use the impulse-momentum principle, which states that the change in momentum of an object is equal to the impulse applied to it.
The impulse (J) exerted on an object can be calculated using the equation:
J = Δp = m * Δv
where m is the mass of the object and Δv is the change in velocity.
In this case, the mass of the baseball (m) is 0.145 kg. The initial velocity of the baseball (u) is 27 m/s horizontally, and the final velocity of the baseball (v) is 0 m/s when it reaches its maximum height. Therefore, the change in velocity is:
Δv = v - u = 0 - 27 = -27 m/s
Next, we need to calculate the change in momentum. Using the equation mentioned earlier:
Δp = m * Δv = 0.145 kg * (-27 m/s) = -3.915 kg*m/s
Since the change in momentum is negative, it means the momentum of the baseball reverses direction when it comes into contact with the bat.
The contact time (Δt) is given as 2.35 milliseconds, which needs to be converted to seconds:
Δt = 2.35 ms = 2.35 * 10^(-3) s
Finally, we can calculate the average force (F) exerted during contact using the equation:
F = Δp / Δt
Plugging in the values:
F = -3.915 kg*m/s / 2.35 * 10^(-3) s
F ≈ -1,668.1 N
Therefore, the average force exerted by the bat on the ball during contact is approximately -1,668.1 Newtons. The negative sign indicates that the force is in the opposite direction to the positive axis, according to the chosen coordinate system.
F average= delta p/delta t
delta p =MV2-MV1