Question

A particle initially at rest starts moving along x - axis .It's acceleration varies with time as a=6t.If it starts from x=10m Then its position at t=2s is?

Answer 1

Given:

Particle starts from rest and moves along the x-axis. Its acceleration varies with time as a = 6t.

It starts from x = 10 m

To find:

Position at t = 2 sec

Calculation:

Acceleration function can be integrated with respect to time in order to get velocity function. The velocity function again has to be integrated in order to get the position or displacement function.

$\therefore \: a = 6t$

$= > \dfrac{dv}{dt} = 6t$

$= > dv = 6t \: dt$

Integrating on both sides ;

$\displaystyle = > \int_{0}^{v} \: dv = \int_{0}^{t} 6t \: dt$

$= > v = 3 {t}^{2}$

$= > \dfrac{dx}{dt} = 3 {t}^{2}$

$= > dx = 3 {t}^{2} \: dt$

Integrating on both sides:

$\displaystyle = > \int_{10}^{x} \: dx = \int_{0}^{2} 3 {t}^{2} \: dt$

$= > x - 10 = \bigg \{ {t}^{3} \bigg \}_{0}^{2}$

$= > x - 10 = 8$

$= > x = 18 \: m$

So , the object will be at 18m at time 2 seconds.