INTRODUCING 5 - days-a-week problem solving session for Math Olympiad and ISI Entrance. Learn More 

May 28, 2017

Projectile Inside a Liquid

Let's discuss a problem useful for Physics Olympiad, based on Projectile Inside a Liquid.

The Problem: Projectile Inside a Liquid

A body of mass m is projected inside a liquid at an angle θ0 with the horizontal at an initial velocity v0. If the liquid develops a velocity-dependent force F= -kv where k is a positive constant, determine the x and y components of the velocity at any instant.

Solution:

A body of mass m is projected inside a liquid at an angle θ0 with the horizontal at an initial velocity v0. The liquid develops a velocity-dependent force F= -kv where k is a positive constant.

Hence,

m dv/dt= -kv

or, dv/dt= -k/m v

or, dv/v= -k/m dt

Integrating both sides,

∫dv/v = -k/m ∫dt

ln|v|= -kt/m+c (where c is a constant of integration)….. (i)

Now, for the x component of velocity,

ln vx=  -kt/m+ c

From the given problem, we have

vx=v0 cosθ0 at t=0

Applying the above condition in eqn.(i), we get

ln (vx/ v0 cosθ0 )= -kt/m

or vx=v0cosθ0e-kt/m

For the y component, we have to consider the acceleration due to gravity g.

Hence,

m dvy/dt= -kvy-mg

or, dvy/(kvy+mg)= -k/m dt

Integrating both sides,

ln|kvy+mg|=-kt/m+c

At t=0, vy=v0sinθ0

Hence,

kvy+mg=(kv0sinθ0+mg) e-kt

 

 

Some Useful Links:

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com