2.5 Constant acceleration (application)

Problem : A particle moves with an initial velocity “ u ” and a constant acceleration “ a ”. What is average velocity in the first “t” seconds?

Solution : The particle is moving with constant acceleration. Since directional relation between velocity and acceleration is not known, the motion can have any dimension. For this reason, we shall be using vector form of equation of motion. Now, the average velocity is given by :

$$\begin{array}{l}{\mathbf{v}}_a=\frac{\Delta \mathbf{r}}{\Delta t}\end{array}$$

The displacement for motion with constant acceleration is given as :

$$\begin{array}{l}\Delta \mathbf{r}=\mathbf{u}t+\frac{1}{2}\mathbf{a}{t}^2\end{array}$$

Thus, average velocity is :

$$\begin{array}{l}{\mathbf{v}}_a=\frac{\Delta \mathbf{r}}{\Delta t}=\frac{\mathbf{u}t+\frac{1}{2}\mathbf{a}{t}^2}{t}=\mathbf{u}+\frac{1}{2}\mathbf{a}t\end{array}$$

Problem : A particle is moving with a velocity $2\mathbf{i}+2t\mathbf{j}$ in m/s. Find (i) acceleration and (ii) displacement at t = 1 s.

Solution : Since velocity is given as a function in “t”, we can find acceleration by differentiating the function with respect to time.

$$\begin{array}{l}\mathbf{a}=\frac{đ}{đt}(2\mathbf{i}+2t\mathbf{j})=2\mathbf{j}\end{array}$$

Thus, acceleration is constant and is directed in y-direction. However, as velocity and acceleration vectors are not along the same direction, the motion is in two dimensions. Since acceleration is constant, we can employ equation of motion for constant acceleration in vector form,

$$\begin{array}{l}\Delta \mathbf{r}=\mathbf{u}t+\frac{1}{2}\mathbf{a}{t}^2\\ \Delta \mathbf{r}=(2\mathbf{i}+2t\mathbf{j})t+\frac{1}{2}x2\mathbf{j}x{t}^2\end{array}$$

For t = 1 s

$$\begin{array}{l}\Delta \mathbf{r}=(2\mathbf{i}+2x1\mathbf{j})x1+\frac{1}{2}x2\mathbf{j}x{1}^2\\ \Delta \mathbf{r}=2\mathbf{i}+3\mathbf{j}\end{array}$$

Note 1 : We should remind ourselves that we obtained displacement using equation of motion for constant acceleration. Had the acceleration been variable, then we would have used integration method to find displacement.

Note 2 : A constant acceleration means that neither its magnitude or direction is changing. Therefore, we may be tempted to think that a constant acceleration is associated with one dimensional motion. As we see in the example, this is not the case. A constant acceleration can be associated with two or three dimensional motion as well. It is the relative directions of acceleration with velocity that determines the dimension of motion – not the dimension of acceleration itself.