Question

a boy starts at rest and slides down a frictionless slide. the bottom of the track is a height h above the ground. the boy then leaves the track horizontally, striking the ground at a distance d. using energy methods calculate the initial hight H of the boy interms of h and d

Answer 1

a boy starts at rest and slides down a frictionless slide. the bottom of the track is a height h above the ground. the boy then leaves the track horizontally, striking the ground at a distance d. using energy methods calculate the initial height H of the boy in terms of h and d

A boy has a potential energy:

$$W _ {p} = m g H$$

which converts to the kinetic energy:

$$W _ {k} = \frac {m v ^ {2}}{2}$$

Initial height:

$$\frac {m v ^ {2}}{2} = m g H \rightarrow H = \frac {v ^ {2}}{2 g}$$

To strike the ground at a distance $d$ :

Time:

$$h = \frac {g t ^ {2}}{2} \rightarrow t = \sqrt {\frac {2 h}{g}}$$

Boy's speed at the bottom of the track:

$$v = \frac {d}{t} = \frac {d}{\sqrt {\frac {2 h}{g}}}$$

Finally:

$$H = \frac {1}{2 g} \left(\frac {d}{\sqrt {\frac {2 h}{g}}}\right) ^ {2} = \frac {d ^ {2}}{4 h}$$

Answer: $H = \frac{d^2}{4h}$ , where $H$ is the initial height of the track.