Question

which graph best represents the relationship between ke and speed?

Explanation:

Step1: Recall the kinetic energy formula

The formula for kinetic energy \(KE=\frac{1}{2}mv^{2}\), where \(m\) is mass (constant for a given object) and \(v\) is speed. This is a quadratic relationship (\(y = ax^{2}\), here \(y = KE\), \(x = v\), \(a=\frac{1}{2}m\)).

Step2: Analyze the graph shapes

• A linear graph (\(y=kx\)) would be for a direct - proportional non - quadratic relationship. The first graph (a straight line) is for a linear relationship (\(KE\propto v\)) which is incorrect as \(KE\propto v^{2}\).
• The third graph (a straight line with negative slope) shows an inverse relationship (\(KE\) decreasing as \(v\) increases) which is wrong since \(KE\) should increase as \(v\) increases.
• The fourth graph (a triangular - like shape) shows a non - monotonic relationship (first increasing then decreasing) which is not in line with \(KE=\frac{1}{2}mv^{2}\) (as \(v\) increases, \(KE\) should keep increasing for \(v\geq0\)).
• The second graph (a curve similar to \(y = x^{2}\) for \(x\geq0\)) is consistent with the quadratic relationship \(KE=\frac{1}{2}mv^{2}\) (when \(v = 0\), \(KE=0\) and as \(v\) increases, \(KE\) increases quadratically).

Answer:

The second graph (the one with a curve that shows \(KE\) increasing with \(v\) in a non - linear, quadratic - like fashion) best represents the relationship between \(KE\) and speed.