Question

The acceleration of a particle starting from rest varies with time according to relation $a=\alpha t+\beta$. What will be the velocity of the particle after time $t$?

Answer 1

Acceleration:

1. The rate of change of velocity is defined as acceleration.
2. It is a vector quantity.

Step 1: Given data

The acceleration of the particle, $a=\alpha t+\beta$

Step 2: Formula used:

We know,

The acceleration of a particle,

$a=\frac{dv}{dt}$

Where $dv$ is change in velocity and $dt$ is change in time.

Step 3: Calculation

$$\begin{gathered} a=\alpha t+\beta \\ \begin{array}{l}\Rightarrow \frac{dv}{dt}=\alpha t+\beta \\ \Rightarrow dv=\left(\alpha t+\beta \right)dt\end{array} \end{gathered}$$

On integrating,

$\begin{array}{l}{\int }_0^{v}dv={\int }_0^t(\alpha t+\beta )dt\\ {\left[v\right]}_0^v={\left[\frac{\alpha t^2}{2}+\beta t+C\right]}_0^t\\ v=\frac{\alpha t^2}{2}+\beta t+C\end{array}$

where $C$ is the constant of integration.

Since a particle starts from rest, its initial velocity is zero.

Therefore when $t=0$, $v=0$.

$$\begin{gathered} v=\frac{\alpha t^2}{2}+\beta t+C \\ \Rightarrow C=0 \end{gathered}$$

Therefore, the velocity of the particle after a time $t$ will be$\begin{array}{l}\frac{\alpha t^2}{2}+\beta t\end{array}$.