A baseball player swings his 2 kg bat with a speed of 15 m/s. They hit a 0.142 kg baseball which was
approaching at a speed of 40 m/s. the ball rebounds in the other direction at 45 m/s.
If the baseball and bat are in contact for 5 ms, what is the average force the bat exerts on the ball?
How does this compare to the gravitational force on the ball?
To find the average force exerted by the bat on the ball, we can use the impulse-momentum principle. The impulse is the change in momentum of an object, and can be calculated using the equation:
Impulse = change in momentum
The change in momentum can be calculated as the difference between the initial momentum and the final momentum:
Change in momentum = final momentum - initial momentum
1. Calculating the initial momentum of the ball:
Momentum = mass * velocity
Initial momentum of the ball = 0.142 kg * (-40 m/s) = -5.68 kg·m/s
2. Calculating the final momentum of the ball:
Final momentum of the ball = 0.142 kg * (45 m/s) = 6.39 kg·m/s
3. Calculating the change in momentum:
Change in momentum = 6.39 kg·m/s - (-5.68 kg·m/s) = 12.07 kg·m/s
Now, we will determine the duration of the contact between the ball and the bat:
Duration of contact = 5 ms = 0.005 s
Using the impulse-momentum principle:
Impulse = average force * duration of contact
4. Calculating the average force:
Average force = impulse / duration of contact
Average force = (12.07 kg·m/s) / (0.005 s) = 2414 N
Therefore, the average force exerted by the bat on the ball is 2414 N.
To compare this to the gravitational force on the ball, we need to calculate the gravitational force acting on the ball:
Gravitational force = mass of the ball * acceleration due to gravity
Gravitational force = 0.142 kg * 9.8 m/s^2 = 1.396 N
So, the average force exerted by the bat on the ball (2414 N) is significantly larger than the gravitational force on the ball (1.396 N).
To find the average force exerted by the bat on the ball, we can use the principle of impulse. Impulse is defined as the change in momentum of an object, and it is given by the product of the force exerted on the object and the time interval over which the force is applied.
1. First, let's calculate the initial momentum of the baseball. The momentum (p) of an object is given by the product of its mass (m) and its velocity (v):
momentum_initial = mass_baseball * velocity_baseball_initial
= 0.142 kg * 40 m/s
= 5.68 kg·m/s
2. Next, let's calculate the final momentum of the baseball after the collision. The final momentum is given by:
momentum_final = mass_baseball * velocity_baseball_final
= 0.142 kg * (-45 m/s)
= -6.39 kg·m/s
(Note: The negative sign indicates that the baseball is moving in the opposite direction after the collision.)
3. The change in momentum of the baseball is given by the difference between the initial momentum and the final momentum:
Δp = momentum_final - momentum_initial
= (-6.39 kg·m/s) - (5.68 kg·m/s)
= -12.07 kg·m/s
4. Now, we can calculate the average force exerted on the baseball using the equation for impulse:
average_force = Δp / Δt
However, it is given that the baseball and bat are in contact for 5 ms, which is equivalent to 0.005 s.
average_force = (-12.07 kg·m/s) / (0.005 s)
= -2414 N
(Note: The negative sign indicates that the force is exerted in the opposite direction of the initial motion of the baseball.)
Therefore, the average force exerted by the bat on the ball is -2414 N.
To compare this with the gravitational force on the ball, we need to calculate the weight of the ball. The weight (W) of an object is given by the product of its mass and the acceleration due to gravity (g):
weight = mass * acceleration_due_to_gravity
Since the mass of the baseball is 0.142 kg and the acceleration due to gravity is approximately 9.8 m/s²,
weight = 0.142 kg * 9.8 m/s²
= 1.3956 N
Therefore, the gravitational force on the ball is approximately 1.396 N.
Comparing the average force exerted by the bat on the ball (-2414 N) with the gravitational force on the ball (1.396 N), we can see that the force exerted by the bat is significantly greater in magnitude.