Finding the constant of this retarding force

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In summary, the conversation discusses how to find the velocity and position of an object in motion, given the mass and initial and final velocities. By using integration and a well-known trick, the velocity and position can be expressed as functions of time and distance, respectively. The constant and time can then be determined using the given data.
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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.

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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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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1. What is the constant of this retarding force?

The constant of a retarding force is a value that represents the strength of the force acting against the motion of an object. It is typically denoted by the letter "k" and is measured in units of Newtons per meter (N/m).

2. How do you find the constant of a retarding force?

The constant of a retarding force can be found by conducting experiments and collecting data on the motion of an object. The constant can then be calculated using the formula k = F/m, where F is the force applied and m is the mass of the object.

3. What factors affect the constant of a retarding force?

The constant of a retarding force can be affected by several factors, including the surface area of the object, the shape and size of the object, and the medium through which the object is moving (such as air or water).

4. Why is it important to know the constant of a retarding force?

Knowing the constant of a retarding force is important in understanding and predicting the motion of an object. It allows scientists to calculate the amount of force acting against the motion of an object and can help in designing more efficient and effective technologies.

5. Can the constant of a retarding force change?

Yes, the constant of a retarding force can change depending on the factors mentioned above. It can also vary with changes in temperature, pressure, and other environmental conditions. It is important for scientists to consider these factors when calculating and using the constant of a retarding force.

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