Question

Aparticle starts from rest at the point x=10 and moves along the x-axis with acceleration

function a(t) = 12t. Find the resulting position function x(t).

Answer 1

Acceleration is the derivative of velocity with respect to time:

$$a(t)=\dfrac{dv}{dt}.$$

We can find the velocity of the particle by integrating the acceleration function over time:

$$\displaystyle\intop_{v_0}^v dv=\displaystyle\intop_{0}^t a(t)dt,$$ $$v-v_0=12\displaystyle\intop_{0}^t tdt,$$ $$v=v_0+6t^2=0+6t^2=6t^2.$$

Velocity is the deriative of position with respect to time:

$$v(t)=\dfrac{dx}{dt}.$$

We can find the position of the particle by integrating the velocity function over time:

$$\displaystyle\intop_{x_0}^x dx=\displaystyle\intop_{0}^t v(t)dt,$$ $$x-x_0=6\displaystyle\intop_{0}^t t^2dt,$$ $$x=x_0+2t^3,$$ $$x=10+2t^3$$