The lob in tennis is an effective tactic when your opponent is near the net. it consists of lifting the ball over his head, forcing him to move quickly away from the net (see the drawing). suppose that you loft the ball with an initial speed of 15.0 m/s at an angle of 50.0° the horizontal. at this instant your opponent is 10.0 m away from the ball. he begins moving away from you 0.22 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?

Minimum average speed with which the opponent must move to hit the ball is 5.56 m/s.

Explanation:

To find the average speed of the opponent, we must find the distance he has to cover and the time he has to cover it. For this we need to find:

a. time at which the ball is at 2.10 m above the launch point

b. distance of ball from the launch point at that time.

We know that

initial velocity = Vi = 15 m/s

Horizontal component of Vi = Vx = Vi*cos(50°) = 9.64 m/s

Vertical Component of Vi = Vy = Vi*sin(50°) = 11.50 m/s

a. Using 2nd equation of motion, we will find time 't' taken by the ball to reach the height 2.10 m

h = Vy*t - 0.5*g*t²

2.1 = 11.5t - 0.5(9.8)t²

4.9t² - 11.5t + 2.1 = 0

Implies that t = 2.15 s or t = 0.20 s

We know that the ball will reach 2.1 m height twice, first while going up(t = 0.20s) and second while coming down(t = 2.15s). Opponent begins moving after time 'T' of hitting the ball where T = 0.22s. As T > (t = 0.20s), it means the opponent will reach the ball when it is coming down and t at that point will be equal to 2.15s.

b. Finding distance 'x' of ball from launch at t = 2.15s:

x = Vx * t

x = 9.64*2.15

x = 20.73 m

Finding the distance 'd' opponent needs to cover to reach the ball at height 2.10 m:

d = (distance of ball at height 2.10 m) - (distance between the two players)

d = 20.73 - 10

d = 10.73 m

Finding time 't°' opponent has to reach the ball:

t° = Time taken by the ball to reach the hitting point - Time player took to decide his move

t° = 2.15 - 0.22s

t° = 1.93 s

Average Speed of the opponent = d/t°

= 10.73/1.93

= 5.56 m/s

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