Question

Given a 500 kg cart which rests at the top of a hill. The cart then rolls down to the bottom of the hill. If the velocity of the cart at the bottom of the hill = 14.0 m/s then what is the height of the hill?

Answer 1

10 m

Step-by-step explanation

Given:

• The mass of the cart, m = 500 kg

,

• The velocity of the cart at the bottom of the hill, v = 14.0 m/s

,

• The velocity of the cart at the top of the hill, v₀ = 0 m/s (we know that it is at rest)

Unknown:

• The height of the hill, h.

By the law of conservation of energy, the initial energy of the cart at the top of the hill must be equal to the final energy at the bottom of the hill. Knowing that the cart starts from rest, we can conclude that when it is at the top it has gravitational potential energy and no kinetic energy - since it is not moving. Also, when it reaches the bottom of the hill, its potential energy drops to 0, while its kinetic energy is at its maximum - which is the same as the potential energy the cart had at the top of the hill,

`\begin{gathered} PE_i=KE_f \\ m\cdot g\cdot h=(1)/(2)\cdot m\cdot v^2 \end{gathered}`

Solving for h,

`h=(v^2)/(2\cdot g)`

As we can see, it does not depend on the mass of the cart. Replace with the values - remember that g = 9.8 m/s²,

`h=(14^2m^2/s^2)/(2\cdot9.8m/s^2)=(196m^2/s^2)/(19.6m/s^2)=10m`

Hence, the height of the hill is 10 meters.