The lob in tennis is an effective tactic when your opponent is near the net. It
ID: 1327959 • Letter: T
Question
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 18.0 m/s, at an angle of 48.5° 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.)
answer in m/s
Explanation / Answer
Given :
initial vertical velocity, vo = 18 m/s
angle, theta = 48.5 degree
initial vertical velocity, voy = vo sin (theta) { eq.1 }
inserting the values in above eq.
voy = (18 m/s) sin (48.50)
voy = 13.4 m/s
And initial horizontal speed is given as, v0x = vo cos (theta) { eq.2 }
vox = (18 m/s) cos (48.50)
vox = 11.9 m/s
using equation of motion 2, y = voy t + (1/2) ay t2 { eq.3 }
rearranging an above eq., we get
t = - voy +/- sqrt [voy2 - 4 (1/2) ay (-y)] / 2 (1/2) ay
t = - voy +/- sqrt [voy2 + 2 ay y] / ay { eq.4 }
where, y = vertical height above its launch point = 2.1 m
ay = acceleration due to gravity downwards = -9.8 m/s2
inserting the values in eq.4,
t = - (13.4 m/s) +/- sqrt [(13.4 m/s)2 + 2 (-9.8 m/s2) (2.1 m)] / (-9.8 m/s2)
t = - (13.4) +/- sqrt (138.4) / (-9.8 m/s2)
t = - (13.4) +/- (11.7) / (-9.8 m/s2)
t = 0.173 sec or t = 2.561 sec
During the 2.561 s, the horizontal distance travelled by the ball is given as ::
x = vx t { eq.5 }
inserting the values in eq.5,
x = (11.9 m/s) (2.561 s)
x = 30.47 m
At this instant , your opponent is 10.0 m away from the ball. it means that the opponent must move (30.47 m - 10 m) = 20.47 m
time taken by the opponent, t = (2.561 s - 0.3 s) = 2.261 sec
At a minimum average speed, the opponent must move which will be given as :
vmin. = distance, x / time, t { eq.6 }
vmin. = (20.47 m) / (2.261 s)
vmin. = 9.05 m/s
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