During the semi-finals of the U. S. Open 2007 Grand Slam Tennis, Venus Williams was in the second set of her match against Justine Henin when her serve was clocked (by a radar gun) at 77 miles per hour. The ball left her racquet horizontally at a height of 2.37 m above the surface of the tennis court. Venus was standing at the service line 12.0 m from the net that has a height of 0.9 m. If the ball clears the net and lands within 7.0 m of the net on the other side, the serve is considered “good" according to the regulations of the game because the ball lands inside the service box on the other side of the net. (See the sketch of a tennis court below.) If the ball clears the net but lands beyond the service box on the other side, this serve is considered a fault. The player receiving the service ball cannot hit the ball before it strikes the court inside the service box. (a) With what minimum speed (in SI units) is Venus required to strike the ball for it to just clear the net? State assumption(s) and support your response quantitatively. (b) Was the serve clocked at 77 miles per hour "good" or was it a fault? If this ball goes over the net, how far will it land from the backline of the service box on Justine’s side of the net. Support your response quantitatively. (c) Sketch graphs of the ball’s displacement, velocity, and acceleration as functions of time (six graphs.) (d) Williams’ slowest serve was at 71 miles per hour. She used the same technique described above but she obviously did not strike this ball as hard. After the match, a local sportscaster said "By serving a slower ball that cleared the net, Williams gave Henin more time to react to her serve because the ball was in the air a little longer than when Williams served her fast ball". Does the sportscaster speak with authority on this subject? Use physics principles to support your response.
(a) To determine the minimum speed required for the ball to just clear the net, we need to consider the height of the net and the distance from the service line to the net.
The net height is given as 0.9 m, and the height at which the ball leaves the racquet is given as 2.37 m. The difference in height is the vertical distance the ball needs to clear:
Vertical distance = 2.37 m - 0.9 m = 1.47 m
Now, let's consider the horizontal distance. The distance from the service line to the net is given as 12.0 m. We know that the ball needs to land within 7.0 m on the other side, so the total horizontal distance the ball needs to cover is:
Total horizontal distance = 12.0 m + 7.0 m = 19.0 m
To find the minimum speed required, we can use the equation for projectile motion:
Horizontal distance = initial horizontal velocity x time
Since we're looking for the minimum speed, we can set the angle of projection to be 45 degrees (which gives us the maximum possible range). The time of flight can be calculated using the vertical motion of the ball:
Vertical distance = initial vertical velocity x time - 0.5 x acceleration x time^2
Since the ball starts and lands at the same height, the initial and final vertical velocities cancel out, and the equation simplifies to:
1.47 m = 0.5 x 9.8 m/s^2 x time^2
Solving for time, we find:
time = sqrt(1.47 m / (0.5 x 9.8 m/s^2))
Plugging this into the equation for horizontal distance, we have:
19.0 m = initial horizontal velocity x sqrt(1.47 m / (0.5 x 9.8 m/s^2))
Solving for the initial horizontal velocity, we find:
initial horizontal velocity = 19.0 m / sqrt(1.47 m / (0.5 x 9.8 m/s^2))
Now we can convert this to SI units. 77 miles per hour is approximately equal to 122.81 km/h. Converting this to m/s, we have:
77 miles/hour x (1 hour / 3600 seconds) x (1000 m / 1 km) = 34.45 m/s
Comparing this value to the calculated minimum speed, we can determine if Venus' serve cleared the net.
(b) To determine if the serve clocked at 77 miles per hour was "good" or a fault, we need to compare the required minimum speed determined in part (a) to the actual speed of the serve.
The actual speed of the serve is given as 77 miles per hour, which we already converted to 34.45 m/s in SI units.
If the actual speed of the serve is greater than or equal to the minimum speed required to clear the net, then the serve is considered "good." Otherwise, it would be a fault.
So, by comparing the actual speed of 34.45 m/s to the minimum speed calculated in part (a), we can determine if Venus' serve was "good" or a fault.
To calculate how far the ball will land from the backline of the service box on Justine's side of the net, we need to consider the horizontal range of the projectile. We can use the same equation as in part (a):
Horizontal distance = initial horizontal velocity x time
Now we can plug in the values we have:
Horizontal distance = 34.45 m/s x time
We already calculated the time in part (a), so we can substitute that value:
Horizontal distance = 34.45 m/s x sqrt(1.47 m / (0.5 x 9.8 m/s^2))
Solving for the horizontal distance, we can find how far the ball will land from the backline of the service box on Justine's side of the net.
(c) To sketch the graphs of the ball's displacement, velocity, and acceleration as functions of time, we need to analyze the projectile motion.
- The displacement graph will be a parabolic curve, where the height above the ground will increase, then decrease as the ball moves across the court. The horizontal axis represents time, and the vertical axis represents the height or displacement of the ball.
- The velocity graph will be a straight line, showing a constant horizontal velocity throughout the motion. The horizontal axis represents time, and the vertical axis represents the horizontal velocity of the ball.
- The acceleration graph will be a horizontal line at zero, indicating that there is no acceleration in the horizontal direction. The horizontal axis represents time, and the vertical axis represents the horizontal acceleration of the ball.
(d) To determine if the sportscaster's statement is correct, we need to analyze the effect of a slower serve on the ball's time in the air.
A slower serve means a smaller initial horizontal velocity. Using the same equations as in part (a) to calculate the time of flight, we can compare the time for the slower serve to the time for the faster serve.
If the time of flight for the slower serve is longer than the time of flight for the faster serve, then the sportscaster's statement would be correct. This would mean that Henin would have more time to react to the slower serve.
By using physics principles and comparing the time of flight for different serves, we can determine if the sportscaster's statement holds true.