A tennis player receives a shot with the ball (0.0600 kg) traveling horizontally at 51.0 m/s and returns the shot with the ball traveling horizontally at 31.0 m/s in the opposite direction. (Take the direction of the ball's final velocity (toward the net) to be the +x-direction.)
(a) What is the impulse delivered to the ball by the racket?
magnitude N · s
direction
(b) What work does the racket do on the ball? (Indicate the direction with the sign of your answer.)
J
The impulse is the change of momentum
0.06 (31 - - 51) = .06(82) toward the net
It does negative work slowing the ball down and then positive work speeding it up.
[force opposite to motion then force in direction of motion]
-(1/2)(.06)(51)^2 +(1/2)(.06)(31)^2 ignoring losses due to compression etc
In other words the ball does work on the racket.
To solve this problem, we need to use the principles of impulse and work. Impulse is a change in momentum, and work is defined as the energy transferred by a force through a displacement.
(a) To find the impulse delivered to the ball by the racket, we need to use the equation:
Impulse = Change in momentum
The momentum of an object is calculated by multiplying its mass (m) by its velocity (v). In this case, we have both the mass and velocities of the ball before and after the shot.
Initial momentum = mass * initial velocity
Final momentum = mass * final velocity
The change in momentum is then given by:
Change in momentum = Final momentum - Initial momentum
Substituting the given values, we can calculate the change in momentum and therefore find the impulse delivered to the ball.
Initial momentum = 0.0600 kg * 51.0 m/s
Final momentum = 0.0600 kg * (-31.0 m/s) (opposite direction)
Change in momentum = Final momentum - Initial momentum
(b) To calculate the work done by the racket on the ball, we can use the equation:
Work = Force * displacement * cos(theta)
Since we have a horizontal shot, the angle between the force applied and the displacement is 0 degrees, and cosine of 0 degrees is 1.
Therefore, the equation simplifies to:
Work = Force * displacement
To find the force, we can use Newton's second law, which states that force is equal to the rate of change of momentum. In this case, we already know the change in momentum (impulse) and the time over which it happens.
Force = Impulse / time
The work done by the racket can then be calculated by multiplying the force by the displacement.
Work = Force * displacement
Now, let's calculate these values.
(a) To find the magnitude of the impulse delivered to the ball:
Initial momentum = 0.0600 kg * 51.0 m/s = 3.06 kg⋅m/s
Final momentum = 0.0600 kg * (-31.0 m/s) = -1.86 kg⋅m/s
Change in momentum = -1.86 kg⋅m/s - 3.06 kg⋅m/s = -4.92 kg⋅m/s (opposite direction)
Therefore, the magnitude of the impulse delivered to the ball is 4.92 N·s.
(b) To find the work done by the racket on the ball:
Force = Change in momentum / time
If we have the value of time, we can calculate the force and then use the force and displacement to find the work. However, if we don't have the time value, we cannot calculate the exact work.
To solve this problem, we need to use the principle of impulse and the work-energy theorem. Let's break it down step by step:
Step 1: Determine the initial momentum of the ball.
The initial momentum of an object is equal to its mass multiplied by its initial velocity. We can calculate the initial momentum using the formula:
p_initial = m * v_initial
Given:
Mass of the ball (m) = 0.0600 kg
Initial velocity (v_initial) = 51.0 m/s
Substituting the values into the formula:
p_initial = 0.0600 kg * 51.0 m/s = 3.06 kg·m/s
Therefore, the initial momentum of the ball is 3.06 kg·m/s.
Step 2: Determine the final momentum of the ball.
The final momentum of an object is equal to its mass multiplied by its final velocity. We can calculate the final momentum using the formula:
p_final = m * v_final
Given:
Mass of the ball (m) = 0.0600 kg
Final velocity (v_final) = -31.0 m/s (opposite direction)
Substituting the values into the formula:
p_final = 0.0600 kg * (-31.0 m/s) = -1.86 kg·m/s
Therefore, the final momentum of the ball is -1.86 kg·m/s.
Step 3: Calculate the change in momentum.
The change in momentum is given by the formula:
∆p = p_final - p_initial
Substituting the calculated values into the formula:
∆p = (-1.86 kg·m/s) - (3.06 kg·m/s) = -4.92 kg·m/s
Therefore, the change in momentum is -4.92 kg·m/s.
Step 4: Calculate the impulse delivered to the ball.
The impulse delivered to the ball is equal to the change in momentum. Impulse is a vector quantity and has both magnitude and direction.
Using the given information, we can now answer the questions:
(a) What is the impulse delivered to the ball by the racket?
Impulse (magnitude) = |∆p| = |-4.92 kg·m/s| = 4.92 kg·m/s
The magnitude of the impulse delivered to the ball by the racket is 4.92 kg·m/s.
(b) What work does the racket do on the ball?
Work is calculated using the formula:
Work = ∆K (Change in kinetic energy)
Since there is no information given about any potential energy changes, we can assume that the initial potential energy and final potential energy are zero. Therefore, the change in kinetic energy (∆K) is equal to the change in the ball's overall mechanical energy.
Substituting the given information into the formula:
Work = ∆K = ∆(Kf - Ki)
= ∆(0.5 * m * (v_final)^2 - 0.5 * m * (v_initial)^2)
= ∆(0.5 * 0.0600 kg * (-31.0 m/s)^2 - 0.5 * 0.0600 kg * (51.0 m/s)^2)
= ∆(-28.755 kg·m^2/s^2 - 79.275 kg·m^2/s^2)
= -108.03 kg·m^2/s^2
Therefore, the work done by the racket on the ball is -108.03 kg·m^2/s^2, indicating that work is done against the motion of the ball.