Question

The position (x) of a body moving along a straight line at time t is given by \[x=3{{t}^{2}}-5t+2\]m.
Find its velocity at
1- t=2 s
2- acceleration at t=2 s and draw the corresponding velocity time graph and acceleration time graph

Answer 1

Hint: The area of a-t graph gives the change in velocity. If the particle starts from rest at any time, the acceleration time graph gives an acceleration of the particle. Putting time value, we can find the value of acceleration at that time. The velocity time graph slope gives acceleration and the area under the velocity time graph gives displacement.

Complete step by step answer:Given position, \[x=3{{t}^{2}}-5t+2\]
To find the velocity we differentiate with respect to time,
\[\dfrac{dx}{dt}=\dfrac{d\{3{{t}^{2}}-5t+2\}}{dt}=6t-5\]
\[\dfrac{dx}{dt}=v=6t-5\]
Now at time, t= 2s,
v= \[6\times 2-5=7m/s\]
now to find the acceleration we differentiate velocity with respect to time,
\[a=\dfrac{dv}{dt}=6m/{{s}^{2}}\]
So, the value of acceleration is constant.
The acceleration time graph will be a straight line parallel to the x axis and touching the y axis at 6

Additional information- when a body moves with a constant speed then it has zero acceleration. When the speed of the body increases then its acceleration is positive and when the speed of the body decreases then acceleration is negative.

Note- We should not get confused that the graph gives us velocity or position, it is only telling us acceleration. If the acceleration time graph is a line parallel to the x-axis it does not mean that the body is at rest but it means the body is moving with constant acceleration.