Question

Given a constant acceleration and assuming linear motion, derive equations for velocity and position of a body with respect to time. Explain what the integration constant represents.

Answer 1

Answer:

v = at + u

$x = ut+\frac{1}{2}at^{2}+x_{0}$

Explanation:

acceleration, a = constant

As we know that acceleration is the rate of change of velocity

$a=\frac{dv}{dt}$

$dv=adt$

integrate on both sides

$\int dv=\int adt$

v = at + u

Where, u is the integrating constant and here it is equal to the initial velocity

Now we know that the rate of change of displacement is called velocity

$v = \frac{dx}{dt}$

$dx=vdt=(u+at) dt$

Integrate on both sides

$\int dx=\int (u+at) dt$

$x = ut+\frac{1}{2}at^{2}+x_{0}$

where, xo is the integrating constant which is initial position of the particle.