A tennis player hits a tennis ball (m = 0.0550 kg) with a force of 32.2 N over a time interval of 0.22 s. If the ball as a velocity of 42.2 m/s before being hit and 87.8 m/s after being hit, how much work did the tennis player do?
Well, the tennis player did some serious work! But not as much as they'd have to do to convince me to play tennis.
To find the work done, we can use the work-energy principle. The work done on an object is equal to the change in its kinetic energy.
First, let's calculate the initial kinetic energy of the ball. Using the equation K = (1/2)mv^2, where m is the mass of the ball and v is its initial velocity, we get:
K1 = (1/2)(0.0550 kg)(42.2 m/s)^2
Now, let's calculate the final kinetic energy of the ball using the same equation, but with the final velocity:
K2 = (1/2)(0.0550 kg)(87.8 m/s)^2
The work done by the tennis player is given by the equation W = ΔK = K2 - K1
So, plug in the values and solve for W:
W = [(1/2)(0.0550 kg)(87.8 m/s)^2] - [(1/2)(0.0550 kg)(42.2 m/s)^2]
W = (1/2)(0.0550 kg)[(87.8 m/s)^2 - (42.2 m/s)^2]
W = (1/2)(0.0550 kg)(7692.84 m^2/s^2 - 1780.84 m^2/s^2)
W = (1/2)(0.0550 kg)(5912 m^2/s^2)
W = 160.9 J
So, the tennis player did 160.9 Joules of work. That's a lot of energy! I hope they had a snack afterwards.
To find the work done by the tennis player, we can use the formula for work:
Work = Force x Displacement
However, we need to find the displacement of the tennis ball first. We can use the equation of motion:
Final velocity^2 = Initial velocity^2 + 2 x acceleration x displacement
Let's start by finding the acceleration using the equation of motion:
Final velocity^2 = Initial velocity^2 + 2 x acceleration x displacement
(87.8 m/s)^2 = (42.2 m/s)^2 + 2 x acceleration x displacement
Simplifying this equation, we get:
7666.84 m^2/s^2 = 1782.84 m^2/s^2 + 2 x acceleration x displacement
Rearranging the equation, we get:
2 x acceleration x displacement = 7666.84 m^2/s^2 - 1782.84 m^2/s^2
2 x acceleration x displacement = 5884 m^2/s^2
Now, let's find the acceleration:
acceleration = (5884 m^2/s^2) / (2 x displacement)
We know that the time interval is 0.22 s, so the displacement can be calculated as:
displacement = initial velocity x time
displacement = (42.2 m/s) x (0.22 s)
Plugging the values into the equation, we get:
displacement = 9.284 m
Now, let's find the acceleration:
acceleration = (5884 m^2/s^2) / (2 x 9.284 m)
acceleration = 317.4 m/s^2
Now that we have the acceleration, we can find the force using Newton's second law:
Force = mass x acceleration
Force = (0.0550 kg) x (317.4 m/s^2)
Force = 17.46 N
Finally, we can calculate the work done by the tennis player:
Work = Force x Displacement
Work = (17.46 N) x (9.284 m)
Work = 162.23 Joules
Therefore, the tennis player did 162.23 Joules of work.
To calculate the work done by the tennis player, we can use the work-energy principle, which states that the work done on an object is equal to the change in its kinetic energy.
The change in kinetic energy can be calculated by finding the difference between the final kinetic energy (KEf) and the initial kinetic energy (KEi).
The formula for kinetic energy is:
KE = (1/2) * mass * velocity^2
Let's calculate the initial and final kinetic energy:
Initial kinetic energy (KEi):
KEi = (1/2) * mass * velocity^2
= (1/2) * 0.0550 kg * (42.2 m/s)^2
Final kinetic energy (KEf):
KEf = (1/2) * mass * velocity^2
= (1/2) * 0.0550 kg * (87.8 m/s)^2
Now, we can calculate the change in kinetic energy (ΔKE):
ΔKE = KEf - KEi
Finally, we can calculate the work done (W) using the work-energy principle:
W = ΔKE
Let's plug in the values and calculate:
Initial kinetic energy (KEi):
KEi = (1/2) * 0.0550 kg * (42.2 m/s)^2
= 47.7963 J (rounded to four decimal places)
Final kinetic energy (KEf):
KEf = (1/2) * 0.0550 kg * (87.8 m/s)^2
= 208.9763 J (rounded to four decimal places)
Change in kinetic energy (ΔKE):
ΔKE = KEf - KEi
= 208.9763 J - 47.7963 J
= 161.1800 J
The work done by the tennis player is therefore 161.1800 Joules.