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
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