# Finding the constant of this retarding force

## Homework Statement:

A block of mass ##100 kg## is moving with velocity ##27,7 \frac{m}{s}##, which reduces to ##15 \frac{m}{s}## after a distance of ##200 m##. This change in velocity is caused by a force ##Fr=-cv^2## where ##c## is a constant and ##v## the velocity.
Find the value of the constant, the time that it takes to move the distance given and an expression for velocity in function of position.

## Relevant Equations:

##-Fr=m.a##
##-Fr=m.a##
##-cv^2=m.a##
##-cv^2=m.\frac{dv}{dt}##
##dt=-\frac{m}{cv^2} dv##

After integrating, I get
##t=\frac{m}{c.v}-\frac{m}{c.v_0}##
Then, solving for ##v## we get
##v=\frac{m.v_0}{v_0.t.c+m}##
##\frac{dx}{dt} = \frac{m.v_0}{v_0.t.c+m}##

After integrating that, I get an expression for ##x(t)##.
But how can I get the constant and the time? Because they are unknowns and if I try to use ##t(v)## and ##x(t)## I get an equation which I can't solve.

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Since you haven't been given the time here, it might be smarter to find the velocity as a function of distance, instead of finding it as a function of time. There is a well-known trick that is frequently used in these kinds of problems:
$$\frac{dv}{dt} = \frac{dv}{dx}\frac{dx}{dt}$$

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Delta2
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If you follow the hint by @Antarres you ll be able to find ##x(v)## which will be much simpler and will allow you to determine the constant c from the data given (200m,27.7m/s,15m/s). Then you can find the time t by the equation you have already found in the OP.

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