A tennis ball of mass 0.057 kg is served. It strikes the ground with a velocity of 54 m/s (120 mph) at an angle of 21° below the horizontal. Just after the bounce it is moving at 51 m/s at an angle of 19° above the horizontal. If the interaction with the ground lasts 0.064 s, what average force did the ground exert on the ball?
Why did the tennis ball bring a calculator to the match? Because it knew there would be a lot of momentum going back and forth!
To find the average force exerted by the ground on the ball, we can use the formula:
Average Force = (Change in Momentum) / (Time)
First, let's find the change in momentum. The initial momentum of the ball is given by:
Initial Momentum = (Mass) * (Initial Velocity)
The final momentum of the ball is given by:
Final Momentum = (Mass) * (Final Velocity)
The change in momentum is then:
Change in Momentum = Final Momentum - Initial Momentum
Now, let's calculate the initial and final momenta of the ball.
Initial Momentum = (0.057 kg) * (54 m/s * cos(21°))
Final Momentum = (0.057 kg) * (51 m/s * cos(19°))
To convert from mph to m/s, we can use the conversion factor:
1 mph = 0.44704 m/s
Now, let's substitute the values into the equations and calculate:
Initial Momentum = 0.057 kg * 54 m/s * cos(21°) ≈ 2.634 kg m/s
Final Momentum = 0.057 kg * 51 m/s * cos(19°) ≈ 2.618 kg m/s
Change in Momentum ≈ 2.618 kg m/s - 2.634 kg m/s ≈ -0.016 kg m/s
Next, we can calculate the average force using the formula mentioned earlier:
Average Force = (Change in Momentum) / (Time)
Average Force ≈ -0.016 kg m/s / 0.064 s ≈ -0.25 N
Now, since force is a vector quantity, the negative sign indicates that the force is in the opposite direction to the initial direction of the ball.
So, the average force exerted by the ground on the ball is approximately 0.25 Newtons, in the opposite direction to the initial motion.
To find the average force exerted by the ground on the ball, 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.
Step 1: Find the initial momentum of the ball.
The initial momentum of the ball can be calculated using the equation:
p_initial = m * v_initial
Given:
Mass of the ball (m) = 0.057 kg
Initial velocity of the ball (v_initial) = 54 m/s
The initial momentum is:
p_initial = 0.057 kg * 54 m/s
Step 2: Find the final momentum of the ball.
The final momentum of the ball can be calculated using the equation:
p_final = m * v_final
Given:
Final velocity of the ball (v_final) = 51 m/s
The final momentum is:
p_final = 0.057 kg * 51 m/s
Step 3: Find the change in momentum.
The change in momentum is given by the equation:
Δp = p_final - p_initial
Substituting the values:
Δp = (0.057 kg * 51 m/s) - (0.057 kg * 54 m/s)
Step 4: Calculate the time of interaction.
The time of interaction, also known as the contact time, is given as 0.064 s.
Step 5: Calculate the average force.
The average force exerted on the ball can be calculated using the equation:
Average Force = Δp / contact time
Substituting the values:
Average Force = Δp / 0.064 s
Now, you can calculate the average force by plugging in the values and solving the equation.
To find the average force exerted by the ground on the tennis ball, 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 can be calculated by:
Impulse = Final momentum - Initial momentum
The initial momentum of the tennis ball can be calculated using its mass and initial velocity:
Initial momentum = mass * initial velocity
The final momentum of the tennis ball can be calculated using its mass and final velocity:
Final momentum = mass * final velocity
Since the initial and final velocities are given as vectors with both magnitude and direction, we need to separate them into horizontal and vertical components.
Horizontal (x) Component:
Initial horizontal velocity = initial velocity * cos(angle below horizontal)
Final horizontal velocity = final velocity * cos(angle above horizontal)
Vertical (y) Component:
Initial vertical velocity = initial velocity * sin(angle below horizontal)
Final vertical velocity = final velocity * sin(angle above horizontal)
Now, we can calculate the initial and final momenta in both the x and y directions:
Initial momentum in x direction = mass * initial horizontal velocity
Initial momentum in y direction = mass * initial vertical velocity
Final momentum in x direction = mass * final horizontal velocity
Final momentum in y direction = mass * final vertical velocity
The change in momentum in the x direction is:
Change in momentum in x direction = Final momentum in x direction - Initial momentum in x direction
Similarly, the change in momentum in the y direction is:
Change in momentum in y direction = Final momentum in y direction - Initial momentum in y direction
The total change in momentum can be calculated by taking the square root of the sum of the squares of the changes in the x and y directions:
Total change in momentum = √((Change in momentum in x direction)^2 + (Change in momentum in y direction)^2)
Finally, to find the average force exerted by the ground, we divide the change in momentum by the duration of the interaction:
Average force = Total change in momentum / Duration of interaction
Plugging in the given values and performing the calculations will give you the average force exerted by the ground on the tennis ball.