The lob in tennis is an effective tactic when your opponent is near the net. It consists of lofting the ball over his head, forcing him to move quickly away from the net (see the drawing). Suppose that you lob the ball with an initial speed of 16.5 m/s, at an angle of 52.5° above the horizontal. At this instant your opponent is 10.0 m away from the ball. He begins moving away from you 0.30 s later, hoping to reach the ball and hit it back at the moment that it is 2.10 m above its launch point. With what minimum average speed must he move? (Ignore the fact that he can stretch, so that his racket can reach the ball before he does.)
first find the time it takes the ball to reach a final height of 2.1m
hf=hi+16.5sin52.5*t-5.9t^2
you know hf-hi=2.1
solve the quadratic for time.
Then how far does the person have to travel?
in the horizontal
d=16.5cos52.5 * t
and the person has to have an average speed of
v=(d-10)/(t-.3)
To determine the minimum average speed at which your opponent must move in order to reach the ball, we need to analyze the motion of the ball and the opponent.
Step 1: Analyzing the motion of the ball
From the given information, we know that the initial speed of the ball is 16.5 m/s and the launch angle is 52.5° above the horizontal. We are interested in the vertical motion of the ball, specifically when it reaches a height of 2.10 m above its launch point.
Using the following kinematic equation for vertical motion:
hf = h₀ + v₀ₙt - (1/2)gt²
Where:
hf is the final height of the ball (2.10 m)
h₀ is the initial height (0 m)
v₀ₙ is the initial vertical component of velocity (v₀ₙ = v₀sinθ)
t is the time taken (unknown)
g is the acceleration due to gravity (approximately 9.8 m/s²)
Rearranging the equation, we can find the time taken for the ball to reach the desired height:
t = (2hf - 2h₀) / g
t = (2 * 2.10 m) / 9.8 m/s²
t ≈ 0.43 s
Therefore, it takes the ball approximately 0.43 seconds to reach a height of 2.10 m.
Step 2: Analyzing the opponent's motion
The opponent starts moving 0.30 seconds after you hit the ball. This means that the time available for the opponent to reach the ball is reduced by 0.30 seconds.
The opponent's horizontal distance to cover is 10.0 m, and the time available is (t - 0.30 s).
To determine the minimum average speed, we divide the horizontal distance by the reduced time available:
Minimum average speed = 10.0 m / (t - 0.30 s)
Substituting the value of t obtained earlier:
Minimum average speed ≈ 10.0 m / (0.43 s - 0.30 s)
Minimum average speed ≈ 10.0 m / 0.13 s
Calculating the minimum average speed:
Minimum average speed ≈ 77.0 m/s
Therefore, your opponent must move with a minimum average speed of approximately 77.0 m/s to reach the ball when it is 2.10 m above its launch point.