Maximum acceleration with simple car

[enerj]

http://build1.net/rand/car.bmp

Homework Statement

Given the variables in the drawing, and coefficient of friction u, what is the maximum possible acceleration if the car is rear wheel drive? Neglect rotational inertia of the wheels.

Homework Equations

Up and right is (+) for forces along with counter-clockwise being (+) for moments
$$\Sigma$$F = m * a
$$\Sigma$$M_c = 0

The Attempt at a Solution

I have established the following variables in my attempt to find the solution;
W - vehicle weight
N_f - normal force on front wheel
N_r - normal force on rear wheel
F_f - frictional force

I gave the car a weight, W, applied downward at the center of mass. From this, a normal force is applied upward at the front and rear wheel, N_f and N_r, respectively.

There is also a frictional force opposing the cars forward motion, F_f pointing to the left.

Summing the forces in the y direction yields W = N_f + N_r

And the moment equation is
$$\Sigma$$M_c = 0 = -(N_r * A) + (N_f * (B - A)) + (F_f * Y)

I am pretty sure that the force the tire applies to the ground will have to be at the verge of breaking friction for maximum acceleration, or u * N_r

Unfortunately I could not relate the previous equations and would appreciate a pointer. By the way, the solution is

[(B - A) g * u ] / [(B - uY)

where g is gravity.

Thanks guys

 

[Delphi51]

Welcome to PF, enerj.
I think the starting point on this question is to ask why wouldn't the "maximum" acceleration be infinite? Is there some limit to the engine power or perhaps to its grip on the road? No use writing any equations until you've got this "grip" on the question.

 

[enerj]

Because as soon as the wheel slips, or breaks friction, the car as a whole will have reached maximum acceleration.

Can I represent this by using F = ma so a = F / m where m is the mass of the car and F is the friction force, so a = u * Nr / m?

 

[Delphi51]

Yes, you've got it! Note that when you fill in the detailed formula for N, the m's will cancel out and you have only a very simple calculation for the acceleration.