Question

A sled slides without friction down a small, ice-covered hill. If the sled starts from rest at the top of the hill, its speed at the bottom is 8.20 m/s. (a) On a second run, the sled starts with a speed of 2.80 m/s at the top. When it reaches the bottom of the hill, is its speed 11.0 m/s, more than 11.0 m/s, or less than 11.0 m/s?

more than 11.0 m/s

equal to 11.0 m/s

less than 11.0 m/s

Explain.

(b) Find the speed of the sled at the bottom of the hill after the second run. m/s

Answer 1

Let h be the height of the hill, the potential energy of the sled at the top of the hill is

where m is the mass of the sled and g is the gravitational acceleration of the Earth. The potential energy at the bottom of the hill is zero.

Let be the velocity of the sled at the top of the hill. The kinetic energy of the sled at the top is

Let be the velocity of the sled at the bottom of the hill. The kinetic energy of the sled at the bottom is

The mechanical energy of the sled at the top of the hill is

The mechanical energy of the sled at the bottom of the hill is

Given the sled slides without friction, the mechanical energy of the sled is conserved

In the first case, we get

In the second case, . Using equation (1)

Since 2gh depends only on the height of the hill and the gravitational acceleration of the Earth, it is the same in both the cases.

(a) The speed is less than 11.0 m/s

(b) The speed in the second run is 8.66 m/s