# To stimulate a roller coaster, a cart is placed at the height of 4 m and allowed to roll from rest to the bottom. Find each of the following for the cart if friction can be ignored: a) the speed at the height of 1 m, b) the height when the speed is 3 m/s?

## Not sure how to do these without a given mass.

Aug 10, 2018

a) 7.67 ms^-1
b) 3.53m

#### Explanation:

As it is said not to consider about frictional force,during this descent,total energy of the system will remain conserved.

So,when the cart was on top of the roller coaster,it was at rest,so at that height of $h = 4 m$ it had only potential energy i.e $m g h = m g 4 = 4 m g$ where, $m$ is the mass of the cart and $g$ is acceleration due to gravity.

Now,when it will be at a height of $h ' = 1 m$ above the ground,it will have some potential energy and some kinetic energy.So,if at that height its velocity is $v$ then total energy at that height will be $m g h ' + \frac{1}{2} m {v}^{2}$

so,we can write,

$m g h = m g h ' + \frac{1}{2} m {v}^{2}$

or, $4 g = g + \frac{1}{2} {v}^{2}$ (see $m$ is getting cancelled out from both side)

Putting, $g = 9.81 m {s}^{-} 2$ we get,

$v = 7.67 m {s}^{-} 1$

Again,using the same equation,if you take $v = 3 m {s}^{-} 1$ then $h ' '$ i.e height at which velocity wil become $3 m {s}^{-} 1$ will be found in this below mentioned way!

$m g h = m g h ' ' + \frac{1}{2} m {\left(3\right)}^{2}$

or, $4 g = h ' ' g + \frac{9}{2}$

or, $h ' ' = 3.53 m$

so, at $3.53 m$ above the ground velocity would have been $3 m {s}^{-} 1$