The motion of a particle is described by the equation u = at . The distance travelled by the particle in the first 4 seconds

The correct option is 8a.

Let's find the distance travelled by the particle step-by-step.

We are given that the velocity (speed) of the particle as a function of time $t$ is given by:
$u=at$
where $a$ is a constant representing acceleration, and $t$ is the time.

The distance $s$ travelled by a particle in a given time interval from $t=0$ to $t=T$ is obtained by integrating the velocity function with respect to time:
$s={\int }_0^{T}udt$

Substituting the expression for velocity $u=at$ into the integral, we get:
$s={\int }_0^{T}atdt$

Since $a$ is a constant, we can pull it out of the integration:
$s=a{\int }_0^{T}tdt$

Integrating $t$ with respect to $t$ yields:
$s=a{\left[\frac{t^2}{2}\right]}_0^T=a(\frac{T^2}{2}-0)=\frac{a{T}^2}{2}$

We are asked to find the distance travelled in the first 4 seconds, so we substitute $T=4$ seconds into our equation:
$s=\frac{a·4^2}{2}$

Calculating the square of 4:
$4^2=16$

Substituting this back into the equation for $s$:
$s=\frac{16a}{2}=8a$

Therefore, the distance travelled by the particle in the first 4 seconds is $8a$.