How far will a pullback car go? Spring and friction coefficients.

[Totally]

Homework Statement

A pullback toy car is pulled back. How far will it go before it stops?

x(0)=0
v(0)=0
a(x<x₀)=-k(x₀-x)
a(x>_x0)=μg
v(t_f)=0
t_f=D

Homework Equations

F_s=-kx
F=mg
F_f=μF

The Attempt at a Solution

d²x/dt²=1/m(k(x₀-x)-μmg)

Later in the run spring constant is irrelevant, so
d²/dt²=1/m(-μmg)



I think I can figure out the μ by pulling the car back on the ramp and lowering the angle between the ramp and the surface until car starts rolling. Then the tangent of that angle tanθ=μ

Would this be correct? Also, what experiment or method would you propose to figure out the coefficient of spring? Disassembling the toy is not an option.

I apologize I didn't put formulas the proper way, however I couldn't get Latex to work even though I followed FAQ instructions.

 

[Sunil Simha]

This is assuming you are letting the car go down the ramp and are slowly increasing the angle of inclination.

I believe that the use of a ramp is correct in finding the coefficient of friction but it should be done without puling back the car. If it were pulled back, the spring force may easily overcome the frictional force at a smaller angle and the reading would be incorrect.

 

[mukundpa]

Is the car sliding on the horizontal surface? If it has rolling wheels then the coefficient of static or kinetic friction becomes irrelevant and rolling friction and air resistance will be the retarding forces. If it is sliding and air resistance is negligible, we can consider kinetic friction and work energy rule will give the results easily.

 

[Totally]

This is assuming you are letting the car go down the ramp and are slowly increasing the angle of inclination.

I believe that the use of a ramp is correct in finding the coefficient of friction but it should be done without puling back the car. If it were pulled back, the spring force may easily overcome the frictional force at a smaller angle and the reading would be incorrect.

I guess I wasn't clear enough, I was thinking about pulling the car back at the lower end of the plane and lowering the angle until it can climb the ramp.

Is the car sliding on the horizontal surface? If it has rolling wheels then the coefficient of static or kinetic friction becomes irrelevant and rolling friction and air resistance will be the retarding forces. If it is sliding and air resistance is negligible, we can consider kinetic friction and work energy rule will give the results easily.

Interesting approach, thanks for the idea! And yes, the toy is being pulled back on the horizontal surface. Do you have any ideas how to figure out the spring coefficient?

 

[Nickie]

I would first clarify the context of this problem. Are we assuming that the pullback car is on a flat surface or on a ramp? This would affect the values of the spring and friction coefficients.

Assuming that the pullback car is on a flat surface, the equation for the acceleration of the car would be:

a = (Fs - Ff)/m

where Fs is the force from the spring, Ff is the force of friction, and m is the mass of the car.

To determine the spring constant, one possible method would be to measure the displacement of the car from its initial position when it comes to a stop. This would give us the value of x0, and we can use the equation Fs = -kx to solve for the spring constant.

Another method would be to measure the time it takes for the car to come to a stop. Using the equation v(tf) = 0, we can calculate the final velocity of the car. Then, using the equation v^2 = v0^2 + 2aΔx, where v0 is the initial velocity (which is 0 in this case), we can solve for the displacement and then use the equation Fs = -kx to solve for the spring constant.

As for determining the coefficient of friction, your proposed method of using the angle of the ramp is a good approach. Another method could be to measure the force required to pull the car back at a constant velocity, and then use the equation Ff = μmg to solve for the coefficient of friction.

In conclusion, there are various methods we can use to determine the spring and friction coefficients in this problem. It is important to carefully consider the context and assumptions of the problem before choosing the most appropriate method.