# 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 $t$
(ii) The distance traveled at any time $t$
(iii) The maximum height attained by the ball

Let $v$ and $h$ be the velocity and height of the ball at any time $t$. Since the time rate of velocity is acceleration, so $\frac{{dv}}{{dt}}$ is the acceleration. Since the ball is thrown upwards, its acceleration is $- g$. Thus, we have

Separating the variables, we have

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

Since $g = 9.8m/{s^2}$, putting this value in (iii), we have

(ii) Since the velocity is the time rate of distance, then $v = \frac{{dh}}{{dt}}$. Putting this value in (iv), we have

Separating the variables of (v), we have

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

(iii) Since the velocity is zero at maximum height, we put $v = 0$ in (iv)

Thus, the maximum height is attained at time $t = 5.1\,\sec$.

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

Thus the maximum height attained is $127.551{\text{m}}$.