# Maximum acceleration with simple car

1. Sep 30, 2009

### enerj

http://build1.net/rand/car.bmp [Broken]

1. The problem statement, all variables and given/known data
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.

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

3. The attempt at a solution
I have established the following variables in my attempt to find the solution;
W - vehicle weight
Nf - normal force on front wheel
Nr - normal force on rear wheel
Ff - 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, Nf and Nr, respectively.

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

Summing the forces in the y direction yields W = Nf + Nr

And the moment equation is
$$\Sigma$$Mc = 0 = -(Nr * A) + (Nf * (B - A)) + (Ff * 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 * Nr

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
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

Last edited by a moderator: May 4, 2017
2. Sep 30, 2009

### 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.

3. Oct 1, 2009

### 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?

4. Oct 1, 2009

### 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.