Question

. dv/dt = a and dx/dt = v(t) where a is constant.Find v(t) in terms of v(0), and a.a.b. Find x(t) in terms of x(0), v(0) and a.

Answer 1

Given:

$\begin{gathered} \frac{dv}{dt}=a \\ \frac{dx}{dt}=v \end{gathered}$

a is a constant

To find:

(a) v(t) in terms of v(0), and a

(b) x(t) in terms of x(0), v(0) and a

Explanation:

(a) We can write,

$dv=adt$

Integrating both sides we can write,

$\begin{gathered} \int_{v(0)}^{v(t)}dv=\int_0^tadt \\ v(t)-v(0)=at \\ v(t)=v(0)+at \end{gathered}$

Hence, the velocity is,

$v(t)=v(0)+at$

(b)

We can also write,

$\begin{gathered} \frac{dx}{dt}=v(t) \\ dx=v(t)dt \end{gathered}$

Integrating both sides we get,

$\begin{gathered} \int_{x(0)}^{x(t)}dx=\int_0^tv(t)dt \\ \int_{x(0)}^{x(t)}dx=\int_0^t[v(0)+at]dt \\ x(t)-x(0)=v(0)t+\frac{1}{2}at^2 \\ x(t)=x(0)+v(0)t+\frac{1}{2}at^2 \end{gathered}$

Hence,

$x(t)=x(0)+v(0)t+\frac{1}{2}at^2$