A projectile of mass m is fired into a liquid at an angle `theta_0` with an initial velocity `v_0` as shown. If the liquid develops a frictional or drag resistance on the projectile which is proportional to its velocity, i. e. `F= -kv` where k is a positive constant, determine the x and y components of its velocity at any instant. Also find the maximum distance `x_(max)` that it travels?

Answer

A, C

Explanation

`a_x = (dv_x)/(dt) = - (kv cos theta)/m = -k/m v_x`
` :. (dv_x)/(v_x) = -k/m dt or int_(v_0 cos theta_0)^(v_x) (dv_x)/(v_x) = - k/m int_0^t dt`
or `v_x = v_0 cos theta_0 e^(-k/m t)` …(i)

Similarly,
`a_y = (dv_y)/(dt) = - (kv sin theta)/m -g = -(k/m v_y + g)`
or `int_(v_0 sin theta_0)^(v_y) (dv_y)/(k/m v_y +g) = - int_0^t dt`
or `m/k [ln(k/m v_y +g)]_(v_0 sin theta_0)^(v_y) = -t`
or `((k/m v_y +g))/((k/m v_0 sin theta_0 + g )) = e^(-k/m t)`
or `v_y =m/k [(k/m v_0 sin theta_0 + g )e^(-k/m t) -g]` ...(ii)
(b) Eq. (i) can be written as
`dx/dt = v_0 cos theta_0 e^(-k/m t)`
or `int_0^x dx = v_0 cos theta_0 int_0^t e^(-k/m t) dt`
or x=(m v_0 cos theta_0)/k [1-e^(-k/m t)]`
or `x_m=(m v_0 cos theta_0)/k`
at `t=oo`