# The Application of Differential Equations in Physics

Differential equations are commonly used in physics problems. In the following example we shall discuss a very simple application of the ordinary differential equation in physics.

__Example__**:**

A ball is thrown vertically upward with a velocity of 50m/sec. Ignoring air resistance, find

**(i)** The velocity of the ball at any time

**(ii)** The distance traveled at any time

**(iii)** The maximum height attained by the ball

Let and be the velocity and height of the ball at any time . Since the time rate of velocity is acceleration, so is the acceleration. Since the ball is thrown upwards, its acceleration is . Thus, we have

Separating the variables, we have

**(i)** Since the initial velocity is 50m/sec, to get the velocity at any time , we have to integrate the left side (ii) from 50 to and its right side is integrated from 0 to as follows:

Since , putting this value in (iii), we have

**(ii)** Since the velocity is the time rate of distance, then . Putting this value in (iv), we have

Separating the variables of (v), we have

In order to find the distance traveled at any time , we integrate the left side of (vi) from 0 to and its right side is integrated from 0 to as follows:

**(iii)** Since the velocity is zero at maximum height, we put in (iv)

Thus, the maximum height is attained at time .

Putting this value of in equation (vii), we have

Thus the maximum height attained is .