Question

a particle moves with a velocity v=at^3 along a straight line. The average speed in time interval t=0 to t=T is​

Answer 1

V=dx/dt. So dx=vdt
By integrating between t=0 to t=T
X= aT^4/4
Average speed =total distance/ total time
Average speed= (aT^4/4)/T= aT^3/4

Answer 2

The average speed in time interval t=0 to t=T is $\dfrac{aT^3}{4}$.

Explanation:

The velocity of a particle is given by :

$v=at^3$

We know that, velocity is given by the rate of change of distance covered as :

$v=\dfrac{dx}{dt}\\\\x=\int\limits {vdt}\\\\x=\int\limits^T_0 {at^3dt}\\\\x=a\dfrac{^4}{4}|^T_0\\\\x=\dfrac{aT^4}{4}$

We know that the average speed is given by total distance divided by total time as :

$v=\dfrac{aT^4}{4T}\\\\v=\dfrac{aT^3}{4}$

So, the average speed in time interval t=0 to t=T is $\dfrac{aT^3}{4}$.

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