Question

A clown in circus juggles with n balls using only one hand. He throws each ball vertically upwards with the same speed at equal time intervals. Denote acceleration of free fall by g.

(a) Find expressions for the speed of projection and height of the i th ball above his hand when he throws the n th ball.
Let he uses n = 4 balls and when he throws the fourth ball, the distance between the second and third ball is d = 50 cm.
(b) Where is the first ball, when the juggler throws the fourth ball?
(c) What is maximum height attained by each ball above the hands of the juggler?

Answer 1

Answer on Question #60032-Physics-Mechanics-Relativity

A clown in circus juggles with n balls using only one hand. He throws each ball vertically upwards with the same speed V at equal time intervals T. Denote acceleration of free fall by g.

(a) Find expressions for the speed of projection and height of the i th ball above his hand when he throws the n th ball.

Let he uses $n = 4$ balls and when he throws the fourth ball, the distance between the second and third ball is $d = 50$ cm.

(b) Where is the first ball, when the juggler throws the fourth ball?

(c) What is maximum height attained by each ball above the hands of the juggler?

Solution

(a)

$$v = V - g t, h = V t - \frac{g t^{2}}{2}.$$ $$t = T (n - i).$$

The total time of flight is

$$\tau = n T.$$

At this time:

$$v = V - g n T = - V$$ $$V = \frac{g n T}{2}$$

Therefore,

$$h = V t - \frac{g t^{2}}{2} = \frac{g n T}{2} T (n - i) - \frac{g}{2} T^{2} (n - i)^{2} = \frac{g T^{2} i (n - i)}{2}$$

(b)

$$h (2) = \frac{g T^{2} 2 (4 - 2)}{2} = 2 g T^{2}$$ $$h (3) = \frac{g T^{2} 3 (4 - 3)}{2} = \frac{3}{2} g T^{2}$$ $$d = 2 g T^{2} - \frac{3}{2} g T^{2} = \frac{1}{2} g T^{2}.$$ $$h (1) = \frac{g T^{2} 1 (4 - 1)}{2} = \frac{3}{2} g T^{2} = 3 d = 150 \text{ cm} = 1.5 \text{ m}.$$

(c)

$$H = h \left(\frac{\tau}{2}\right) = h \left(\frac{n T}{2}\right) = h \left(\frac{4 T}{2}\right) = h (2 T) = \frac{g 4 T}{2} 2 T - \frac{g}{2} (2 T)^{2} = 2 g T^{2}$$ $$H = 4d = 2\,m.$$

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