Question

Ezra (m = 20.0 kg) has a tire swing and wants to swing as high as possible. He thinks that his best option is to run as fast as he can and jump onto the tire at full speed. The tire has a mass of 10.0 kg and hangs 3.50 m straight down from a tree branch. Ezra stands back 10.0 m and accelerates to a speed of 3.62 m/s before jumping onto the tire swing. (a) How fast are Ezra and the tire moving immediately after he jumps onto the swing? m/s (b) How high does the tire travel above its initial height?

Answer 1

a)
`v=5.6725\,m.s^(-1)`

b)
`h= 1.6420\,m`

Step-by-step explanation:

Given:

• mass of the body,
  `M=20\,kg`

• mass of the tyre,
  `m=10\,kg`

• length of hanging of tyre,
  `l=3.5m`

• distance run by the body,
  `d=10m`

• acceleration of the body,
  `a=3.62m.s^(-2)`

(a)

Using the equation of motion :

`v^2=u^2+2a.d`..............................(1)

where:

v=final velocity of the body

u=initial velocity of the body

here, since the body starts from rest state:

`u=0m.s^(-1)`

putting the values in eq. (1)

`v^2=0^2+2* 3.62 * 10`

`v=8.5088\,m.s^(-1)`

Now, the momentum of the body just before the jump onto the tyre will be:

`p=M.v`

`p=20* 8.5088`

`p=170.1764\,kg.m.s^(-1)`

Now using the conservation on momentum, the momentum just before climbing on the tyre will be equal to the momentum just after climbing on it.

`(M+m)* v'=p`

`(20+10)* v'=170.1764`

`v'=5.6725\,m.s^(-1)`

(b)

Now, from the case of a swinging pendulum we know that the kinetic energy which is maximum at the vertical position of the pendulum gets completely converted into the potential energy at the maximum height.

So,

`(1)/(2) (M+m).v'^2=(M+m).g.h`

`(1)/(2) (20+10)* 5.6725^2=(20+10)* 9.8* h`

`h\approx 1.6420\,m`

above the normal hanging position.