### Video Transcript

A body was projected up a rough inclined plane from its bottom. Its initial kinetic energy was 242 joules. The body continued moving until it reached its maximum height and then slid back down to the bottom. When it reached the bottom, its kinetic energy was 186 joules. Find the work done against friction ๐ during the ascent and the gain in gravitational potential energy ๐ when the body was at its maximum height.

Letโs begin by sketching a diagram of the situation. We are told that the body starts at the base or bottom of the plane. It then ascends up the slope until it reaches its maximum height. The body then slides back down the plane so it returns to the bottom. In order to answer this question, we need to recall the conservation of energy. This states that energy is neither created nor destroyed; it only moves from one type to another. This means that the initial energy must be equal to the final energy.

We are told that the initial kinetic energy was 242 joules and the final kinetic energy was 186 joules. As the body starts and finishes at the same point, the gravitational potential energy at this point is equal to zero. This seems to suggest that we have lost energy as we started with 242 joules and we now have 186 joules. The reason for this is that we are told that the plane was rough. This means that we will lose energy due to friction.

We can calculate this work done against friction by firstly subtracting 186 from 242. This is equal to 56. Therefore, the work done against friction during the entire journey is equal to 56 joules. We are only interested in the work done during the ascent. This means that we need to divide 56 by two, as the work done going up the slope will be equal to the work done coming back down. The work done against friction ๐ during the ascent is, therefore, equal to 28 joules.

The second part of our question wants us to calculate the gravitational potential energy ๐ when the body is at its maximum height. There was 242 joules of kinetic energy at the start of the motion. We have just found out that 28 joules was lost during the ascent due to the work done against friction. Subtracting 28 from 242 gives us 214. This means that the gravitational potential energy ๐ when the body was at its maximum height is equal to 214 joules.

On the descent, 28 joules will once again be used for the work done against friction. As 214 minus 28 gives us 186, this ties in with the kinetic energy at the end. The two answers to this question are ๐ is equal to 28 joules and ๐ is equal to 214 joules.