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 v_x=  -kt/m+ c

From the given problem, we have

v_x=v₀ cosθ₀ at t=0

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

ln (v_x/ v₀ cosθ₀ )= -kt/m

or v_x=v₀cosθ₀e^-kt/m

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

Hence,

m dv_y/dt= -kv_y-mg

or, dv_y/(kv_y+mg)= -k/m dt

Integrating both sides,

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

At t=0, v_y=v₀sinθ₀

Hence,

kv_y+mg=(kv₀sinθ₀+mg) e^-kt

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