Question

based on this graph, which chart represents the energy for point a?
height (m) vs. time (s)
graph of height (m) on y - axis (0 - 8) and time (s) on x - axis (0 - 4). the line starts at (0,6), rises to (0.5,8), then falls to (2,2) point a, then to (3,0), then rises to (4,6).
three bar charts:

1. first: ( e_p ) bar height 2, ( e_k ) bar height 6.
2. second: ( e_p ) bar height 6, ( e_k ) bar height 2.
3. third: ( e_p ) bar height 2, ( e_k ) bar height 0 (or very low).

Explanation:

Step1: Analyze the height - energy relation

In a mechanical system (like a projectile or an object in free - fall - like motion here), gravitational potential energy \(E_p=mgh\) (where \(m\) is mass, \(g\) is acceleration due to gravity, \(h\) is height) and kinetic energy \(E_k\). Assuming mass is constant, potential energy is proportional to height. At the initial point (time = 0), height \(h = 6\) m. The total mechanical energy \(E = E_p+E_k\). At the initial moment, if we assume the object is moving upwards, maybe initial kinetic energy is such that total energy is conserved (assuming no air resistance, mechanical energy is conserved). So total energy \(E\) at any point should be equal to the initial total energy. At initial time (\(t = 0\)), \(h = 6\) m. Let's assume at \(t = 0\), if the object starts with some kinetic energy, but actually, looking at the height - time graph, at \(t = 0\), height \(h = 6\) m, and then it goes up to \(h = 8\) m (so it has kinetic energy to move up), then comes down. But for conservation of mechanical energy, \(E_p+E_k=\text{constant}\). At point A, height \(h = 2\) m. So \(E_p\) at A is proportional to \(h = 2\), and \(E_k\) should be such that \(E_p+E_k=\text{constant}\). At the initial point (\(t = 0\)), if we consider when \(h = 6\) m, maybe at that point, if we assume the total energy is such that when \(h = 6\), and then when \(h = 2\), \(E_p\) is 2 (proportional) and \(E_k\) is 6 (since \(2 + 6=8\)? Wait, no, maybe the total energy is equal to the maximum potential energy? Wait, the maximum height is \(h = 8\) m, so maximum \(E_p\) is at \(h = 8\), so total energy \(E = mgh_{max}\). At point A, \(h = 2\) m, so \(E_p=mgh = 2k\) (where \(k = mg\)), and \(E_k=E - E_p=mgh_{max}-mgh=k(8 - 2)=6k\). Wait, no, maybe the initial height is \(h = 6\) m, and then it goes up to \(h = 8\) m (so it has kinetic energy to climb from 6 to 8, so initial \(E_p = 6k\), initial \(E_k\) is enough to take it to 8, so total energy \(E=8k\)). Then at point A, \(h = 2\) m, so \(E_p = 2k\), \(E_k=E - E_p=8k - 2k = 6k\). So \(E_p = 2\), \(E_k = 6\). So the bar graph should have \(E_p = 2\) and \(E_k = 6\), which is the first chart (the one with \(E_p = 2\) and \(E_k = 6\)).

Step2: Match with the options

Looking at the three options:

• First option: \(E_p = 2\), \(E_k = 6\)
• Second option: \(E_p = 6\), \(E_k = 2\)
• Third option: \(E_p = 2\), \(E_k = 0\) (no, that can't be, since it's moving)

Since at point A, the object is in motion (it's on the way down or up? Wait, the height - time graph: after \(t = 1\), it's coming down from \(h = 8\) to \(h = 0\) (at \(t = 3\)), then going up again. At \(t = 2\), point A is at \(h = 2\) m, so it's moving down, so it has kinetic energy. So \(E_p\) is proportional to height (2), \(E_k\) is larger (6) because total energy is conserved (assuming no energy loss). So the first chart (with \(E_p = 2\) and \(E_k = 6\)) is correct.

Answer:

The first bar chart (with \(E_p\) bar of height 2 and \(E_k\) bar of height 6)