A projectile of mass m is shot straight up with initial velocity v0 and feels a

Question

A projectile of mass m is shot straight up with initial velocity v0 and feels a quadratic air resistance force of the form F=-kv^2,where k=mg/u^2.(k and u are constants and g is the acceleration of gravity.)

a)Find the speed of the projectile at time t while the projectile is still moving upwards. (Hint: you will need to know or calculate an indefinite integral and apply initia lconditions.)

b)Take the limit u approaches infinity (that is,k approaches 0)inthe above expression and show thatitgoes over to the standard expression for resistance-free vertical motion.

c)Find the time T at which the projectile reaches its maximum height.(Hint:the result willbeof the form of a constant multiplying the tangent of another constant.)

Explanation / Answer

a) so

- m g - k v^2 = m dv/dt

- m g - mg v^2/u^2 = m dv/dt

dv/dt = - g - g v^2/u^2

dv/(1 + v^2/u^2) = - g dt

integrate both sides

u arctan(v/u) = - gt + C

arctan(v/u) = - gt/u + C

v/u = tan(-gt/u + C)

v = u tan(-gt/u + C)

initial condition t = 0 v = v0

v0 = u tan( C)

C = arctan(v0/u)

v = u tan( -gt/u + arctan(v0/u) )

b) so arctan(v0/u) -> v0/u

tan(-gt/u +v0/u)-> -gt/u + v0/u

so v = u*(-gt/u + v0/u) = -gt + v0

c)

max height when v = 0

tan(-gt/u + arctan(v0/u) ) = 0

-gt/u + arctan(v0/u) = 0
t = u arctan(v0/u)/g

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