# Object of Mass question

• Mar 8th 2012, 01:18 PM
NFS1
Object of Mass question
Hi, I have a past exam questions that I need help with solving. I am having some difficulty on understanding where to start and how to solve the problem. Any help will be appreciated. Below is the question:

1) An object of mass
M is projected vertically upwards from the surface of the Earth. It experiences air resistance which is proportional to the square of the magnitude of its velocity. Newton's equation of motion in this case is

dv/dt
= -g - (B(beta)/M)/v^2

where B (beta)> 0 is the strength of the air resistance and g > 0 is the local gravitational constant.

a) Seperate variables in equation (1) and substitute

u = v(Square root)(B(beta)/Mg)

to obtain an expression for the upward velocity function
v(t), with initialcondition v(0) = v0.

b) Use your answer to the previous part of this question to obtain the verticalheight function,
y(t), giving the height of the object at time t, assume the
initial condition is y(0) = h.

c) Now perform a similar analysis for the case when the object is projecteddownwards to nd that the height function
y(t) is given the expression

y(t) = y0 - M/k log (cosh (A-t)(Square root)(kg/M))/cosh(B)

where
A;B are constants (what is y0?). If the initial velocity is zero, describe the downward velocity as t -> infiniti.

• Mar 9th 2012, 06:01 PM
HallsofIvy
Re: Object of Mass question
I don't see how we can help if we don't know what your difficulty is! You say you don't know where to start but the problem TELLS you what to do: "Separate variables in equation (1) and integrate". If you move every "v" to the left side of the equation and every "t" (there is only dt) to the right, what do you get? What is the integral of both sides?