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.
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 .