Question

At time t=0 a car is moving 6m/s then smoothly accelerate so that the acceleration after t seconds is a a(t)=3t m/s

Answer 1

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

$$dv=adt$$

$$\int dv=\int adt$$ $$v=\int adt$$

Given $a=3t$

$$v=\int(3t)dt$$

$$v(t)=\dfrac{3}{2}t^2+C_1$$

If $v(0)=6$

$$v(0)=\dfrac{3}{2}(0)^2+C_1=>C_1=6$$

$$v(t)=\dfrac{3}{2}t^2+6$$

$$v(t)=\dfrac{ds}{dt}$$

$$ds=vdt$$

$$\int ds=\int vdt$$ $$s=\int vdt$$ $$s=\int(\dfrac{3}{2}t^2+6)dt$$

$$s(t)=\dfrac{1}{2}t^3+6t+s(0)$$

If $s(0)=0$

$$s(t)=\dfrac{1}{2}t^3+6t$$