Question

with what velocity must a ball be thrown vertically upward in order to rise a height of 50m? how long will it be in the air?

Answer 1

With what velocity must a ball be thrown vertically upward in order to rise a height of $50\mathrm{m}$ ? How long will it be in the air?

Solution:

The equation of motion for the ball:

$$H = V t - \frac {g t ^ {2}}{2} (1),$$

$t - \text{flight time to reach the height } H = 50m$

Rate equation for the ball, in the end of the flight speed of the ball is zero:

$$0 = V - g t$$ $$t = \frac {V}{g} (2)$$ $$(2) \text{in} (1): H = V \frac {V}{g} - \frac {g \left(\frac {V}{g}\right) ^ {2}}{2}$$ $$H = \frac {V ^ {2}}{2 g}$$ $$V = \sqrt {2 g H} = \sqrt {2 * 1 0 \frac {m}{s ^ {2}} * 5 0 m} = 3 1. 6 \frac {m}{s}$$

Residence time in the air: time of flight of the ball upward equals time of the fall:

$$T = t _ {u p} + t _ {d o w n} = 2 t = \frac {2 V}{g} = \frac {2 * 3 1 . 6 \frac {m}{s}}{1 0 \frac {m}{s ^ {2}}} = 6. 3 2 s$$

Answer: velocity: $V = 31.6 \frac{m}{s}$

Time in the air: $T = 6.32 \, \text{s}$