When sound travels through air, the air particles vibrate. A graph of displacement against time for a single air particle is shown.

Question 24

When sound travels through air, the air particles vibrate. A graph of displacement against time for a single air particle is shown.

Which graph best shows how the kinetic energy of the air particle varies with time?

Reference: Past Exam Paper – November 2014 Paper 11 & 12 Q23

Solution:

Answer: D.

Kinetic energy = ½ mv²

A graph of displacement against time for a single air particle is shown. 

The gradient of the displacement-time graph gives the velocity of the air particle at that point in time. This is done by calculating the gradient of the tangent at that point.

The gradient (and hence, velocity) is found to be zero at the maximum displacement (the tangent is horizontal) and maximum when the displacement is zero (the tangent is steepest).

Note that the velocity is NOT zero when the displacement is zero. At the points where displacement = 0, we need to draw a tangent to obtain the velocity. It can clearly be observed then that the line is steepest at these points – that is, the velocity is greatest when displacement = 0.

Thus, at time = 0, T and 2T the velocity is zero and hence kinetic energy is zero. [A and C incorrect]

But, between time = 0 and T or between time = T and 2T the displacement is zero (in case) twice. So, the velocity (and kinetic energy) reaches its maximum value 2 times in each of the 2 intervals. [B is incorrect]