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Document Title
Fundamentals of Stress and Vibration 2. Engineering Mechanics Chapter
[A Practical guide for aspiring Designers / Analysts]

Example 1: a ball is thrown into the water, with an initial velocity U (as shown in [Fig 2.16]).
0
The drag experienced by the ball in water is proportional to square of the instantaneous velocity.
Find the time taken by the ball, to travel a depth of ‘d’, where the velocity of the ball becomes v .
0

[Fig 2.16: ball travelling in water]

Solution: the factors affecting the motion of the ball are:

Ø Drag due to water
Ø Gravity

Here, we neglect viscous drag. Therefore, the acceleration of the ball at any instant is given by:

net force gravity force − drag force
Instantaneous acceleration (a) = =
mass of ball mass of ball
The drag force is given by kv , where, ‘k’ is the constant of proportionality. Therefore, we have:
2
mg − kv 2 kv 2
a = = g −
m m

Writing acceleration in terms of velocity, we get:

dv kv 2
= g − - - - - (2.19)
dt m

Equation (2.19) represents the motion of the ball and its instantaneous acceleration. We would
use this equation to find the time of journey, as, we know the initial and final velocities.

QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries, P
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