Question

a skateboarder heads down a smooth, steep course. what graph illustrates the change in kinetic and potential energy during the process? options: a, b, c, d (graphs: a/b with horizontal lines; c/d with curved lines intersecting; labels: kinetic energy (solid), potential energy (dashed))

Explanation:

Step1: Recall Energy Conversion

When a skateboarder moves down a smooth (no friction) ramp, gravitational potential energy (PE) converts to kinetic energy (KE). PE depends on height ($PE = mgh$), so as height decreases (moving down), PE decreases. KE depends on speed ($KE=\frac{1}{2}mv^2$), so as speed increases (moving down), KE increases.

Step2: Analyze Graphs

• Graph A/B: Show constant energy, which is wrong (PE should decrease, KE increase).
• Graph C: KE (solid) increases (curving up, speed increasing more? Wait, no—wait, smooth ramp, so mechanical energy is conserved. PE decreases (exponential? No, height vs time: if moving down a ramp, height decreases linearly? Wait, no, the key is PE and KE should be inversely related, with PE decreasing and KE increasing, and their sum constant. So one curve should be decreasing (PE, dashed) and one increasing (KE, solid), crossing, and their shapes: since $PE = mgh$ and if moving with acceleration (down ramp, constant acceleration? Wait, smooth ramp, so acceleration is constant? Wait, no, the graph of KE vs time: if speed increases linearly (constant acceleration), then KE is proportional to $v^2$, so KE vs time would be a quadratic (parabola) opening up. PE is $mgh$, and height $h$ as a function of time: if moving with constant acceleration, $h$ decreases as a quadratic (since $s = ut+\frac{1}{2}at^2$, so height decreases with $t^2$), so PE is a quadratic opening down. Wait, looking at the graphs:

Graph D: KE (solid) increases (parabola up), PE (dashed) decreases (parabola down), crossing. That matches: as time increases, KE increases (speed up) and PE decreases (height down), with their sum constant (mechanical energy conserved, smooth ramp so no energy loss).

Graph C: KE (solid) starts low, increases, PE (dashed) starts high, decreases, but the shapes? Wait, no—wait the labels: in the problem, the graphs have KE (solid) and PE (dashed). Wait, the correct graph should have KE increasing (as speed increases) and PE decreasing (as height decreases), with KE being a curve that opens up (since $KE \propto v^2$, and $v$ increases linearly with time if acceleration is constant, so $KE \propto t^2$) and PE being a curve that opens down (since $PE \propto h$, and $h$ decreases as $t^2$). So Graph D shows KE (solid) curving up, PE (dashed) curving down, crossing. That's correct.

Answer:

D