Question

a particle moves with a speed v=kt2 where k is a constant t is time ,along a straight line.the total change in its speed in time t = 0 to t =T is

Answer 1

Given info : a particle moves with a pseed v = kt² , where k is a constant , t is time along straight line.

To find  : the total change in its speed in time, t = 0  to t = T

solution : total change in the speed in a given time interval = average speed in that time interval

for variable function of speed,

the average speed is given by, $v_{\text{av}}=\frac{\int v(t),dt}{\int dt}$

here v(t) = kt²

∴ $v_{av}=\frac{\int\limits^T_0{kt^2},dt}{\int\limits^T_0dt}$

= $\frac{k\left[\frac{t^3}{3}\right]^T_0}{T}$

= $\frac{kT^2}{3}$

therefore the total change in its speed in time t = 0 to t = T