A tennis ball of mass 0.0609 kg is served. It strikes the ground with a velocity of 52.5 m/s (117 mi/h) at an angle of 22.0° below the horizontal. Just after the bounce it is moving at 50.5 m/s at an angle of 17.7° above the horizontal. If the interaction with the ground lasts 0.0565 s, what average force did the ground exert on the ball?
To find the average force that the ground exerted on the ball, we will use the impulse-momentum theorem. This theorem states that the change in momentum of an object is equal to the net external force applied to it multiplied by the time over which the force is applied.
The change in momentum of the ball can be calculated using the initial and final velocities of the ball. The momentum of an object is equal to its mass multiplied by its velocity. So, the change in momentum (Δp) is given by:
Δp = m * (vf - vi)
Where:
m = mass of the ball (0.0609 kg)
vf = final velocity of the ball after the bounce
vi = initial velocity of the ball before the bounce
To find the final velocity (vf) and initial velocity (vi) of the ball after and before the bounce respectively, we need to break down the velocities into their horizontal and vertical components.
Given:
Initial velocity (vi) = 52.5 m/s at an angle of 22.0° below the horizontal
Final velocity (vf) = 50.5 m/s at an angle of 17.7° above the horizontal
To find the horizontal and vertical components of velocity, we can use trigonometry. The horizontal component of velocity (vix/vfx) is given by:
vix/vfx = v * cos(θ)
And the vertical component of velocity (viy/vfy) is given by:
viy/vfy = v * sin(θ)
Where:
v = magnitude of velocity (52.5 m/s or 50.5 m/s)
θ = angle (22.0° or 17.7°)
Now, let's calculate the horizontal and vertical components of velocity:
Initial horizontal component of velocity (vix) = vi * cos(θ)
Initial vertical component of velocity (viy) = vi * sin(θ)
Final horizontal component of velocity (vfx) = vf * cos(θ)
Final vertical component of velocity (vfy) = vf * sin(θ)
Using the given values, we can calculate the horizontal and vertical components of velocity.
Next, we can calculate the change in momentum (Δp) using the initial and final velocities:
Δp = m * (vfx - vix)
Now, we need to convert the given time of interaction with the ground to an average force. The average force (Favg) is given by:
Favg = Δp / Δt
Where:
Δt = time of interaction with the ground (0.0565 s)
Finally, substitute the values into the equation to find the average force (Favg).