Question

the following graph models the child-earth system’s kinetic energy (k) and gravitational potential energy (u_g) when the child is at the position shown in the image.
bar graph: energy (j) on y-axis (0, 100, 200, 300, 400, 500); x-axis labels k (green bar at 400 j) and u_g (blue bar at 100 j)
which graph could represent the energy of the child-earth system when the child reaches the surface of the water?
assume frictional forces are small enough to ignore.

Explanation:

Step1: Recall Energy Conservation Principle

In a system with negligible friction (like the child - Earth system here), the total mechanical energy \( E = K+U_{g}\) is conserved. First, calculate the total mechanical energy initially. From the given graph, \( K = 400\space J\) and \( U_{g}=100\space J\), so the total energy \( E=400 + 100=500\space J\).

Step2: Analyze Energy at Water Surface

When the child reaches the surface of the water, the gravitational potential energy \( U_{g}\) will decrease (since height is lower) and the kinetic energy \( K\) will increase, but the total mechanical energy \( E = K + U_{g}\) should still be equal to \( 500\space J\) (because friction is ignored, so mechanical energy is conserved). So we need to find a graph where the sum of the kinetic energy and gravitational potential energy bars is \( 500\space J\), with \( U_{g}\) being smaller than \( 100\space J\) (or \( K\) being larger than \( 400\space J\)) compared to the initial state, but their sum remains \( 500\space J\).

For example, if at the water surface, suppose \( U_{g}\) becomes \( 0\space J\) (if the water surface is taken as the reference point for potential energy), then \( K=500 - 0 = 500\space J\). Or if \( U_{g}\) is some positive value less than \( 100\space J\), \( K\) will be \( 500 - U_{g}\), which will be greater than \( 400\space J\).

Answer:

The graph should have a total mechanical energy (sum of \( K\) and \( U_{g}\)) of \( 500\space J\), with \( U_{g}\) less than \( 100\space J\) (or \( K\) greater than \( 400\space J\)) compared to the initial graph, while \( K + U_{g}=500\space J\). For instance, a graph with \( K = 500\space J\) and \( U_{g}=0\space J\) (or other combinations where their sum is \( 500\space J\)) would be correct.