Question

A particle moves along x-axis with an initial speed $v_0$ = $5m{s}^{-1}$. If its acceleration varies with time as shown in a-t graph in the figure.
a. Find the velocity of the particle at t = 4s.
b. Find the time when the particle starts moving along -x direction.

Answer 1

The velocity of the particle at t = 4s can be given as
${\overrightarrow{v}}_4={\overrightarrow{v}}_0+\Delta \overrightarrow{v}$ ...(i)
where $\Delta \overrightarrow{v}$ = A

(= area under a-t graph during first four seconds)

Referring to a-t graph, we have ...(ii)
where $A_1$ = 5 x 1 = 5,$A_2$ = (1/2) x $x$ x 5,

$A_3$ = (1/2) x (1 - $x$) x 10, and $A_4$ = (1/2) x 2 x 10 = 10

We can find x as following:

Using properties of similar triangles, we have $\frac{x}{5}=\frac{1-x}{10}$
This yields x = 1/3.

Substituting x = (1/3) in $A_2$ and $A_3$
we have $A_2$ = 5/6 and $A_3$ = 10/3.

Then substituting $A_1,A_2,A_3$ and $A_4$ in (ii), we have A = -7.5.

Negative area tells us that change in velocity is along -x direction
$\Delta \overrightarrow{v}$ = -7.5 m/s

Hence substituting in (i),$\Delta {\overrightarrow{v}}_0$ = 5 m/s and $\Delta \overrightarrow{v}$ = -7.5 m/s, we have $\Delta {\overrightarrow{v}}_4$ = ${\overrightarrow{v}}_0$ + $\Delta \overrightarrow{v}$ = 5- 7.5 = -2.5 m/s.