Stopping Distance

Date: 02/25/97 at 18:50:59
From: Benjamin Marshan
Subject: Physics: momentum

A 14700N car is traveling at 25m/s. The brakes are suddenly applied 
and the car slides to a stop. The average braking force between the 
tires and the road is 7100N. How far will the car slide once the 
brakes are applied?

All we think we know is that we should be using p = mv.  Is this 
correct?

Thank You!

Date: 03/07/97 at 18:38:57
From: Doctor Barney
Subject: Re: Physics: momentum

Instead of giving you the answer, how about I put you on the right 
track? (Or the right "road", since this is a car problem.)  Please 
write back again if you need more help.  

Actually, I do not believe that there is enough information given to 
solve the problem, since it says "average" force.  If there is a 
greater force applied earlier in the braking, the car will stop in a 
shorter distance.  If there is a greater force applied later in the 
braking, even if the time average of the force is the same and the 
total time it takes to stop is the same, the car will have traveled a 
longer distance.  Let's assume that the braking force is a "constant" 
7100N.

I can think of two methods to solve this problem, but neither of them 
involves momentum, although there may be other methods.

1) Use Newton's second law, F = ma.  You know the force and you can 
find the mass, so you can calculate a constant acceleration (or 
deceleration, in this case).  From the acceleration you can calculate 
velocity as a function of time.  Next integrate the velocity from time 
0 (when the brakes are applied) to the time when the car stops moving 
to find the distance traveled.  This is an easy integral since the 
velocity function is a straight line when the braking force is a 
constant.  

2) Use energy methods.  The initial kinetic energy of the car will be 
equal to the "work" the car does on the tires/road.  This equation 
looks like 1/2mv^2 = Fd where m is the mass of the car, v is the 
initial velocity, F is the force applied, and d is the distance over 
which the constant force is applied, which is what you are attempting 
to determine.  Have you covered this yet?

Note: With either of these methods, be sure to calculate the mass of 
the car first, since the problem statement gives you the weight of the 
car.
 
-Doctor Barney,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 03/07/97 at 18:49:44
From: Doctor Luis
Subject: Re: Physics:- momentum

You don't really need to consider momentum in this problem. To solve 
it, let's look at the given information.

The weight of the car is 14700N. We can obtain the mass from this:

    w = 14700 N      w = mg         m = 14700 N / 9.8 m/s^2
    g = 9.8 m/s^2    m = w/g          = 1500 kg

The average braking force between the tires and the road is 7100 N. We 
can obtain the deceleration of the car from this:

    f = 7100 N         f = ma      a = 7100 N / 1500 kg
    m = 1500 kg        a = f/m       = 4.73 m/s^2

The initial velocity of the car (before the brakes are applied) is 
25 m/s. Since we know the acceleration of the car and its final 
velocity (0 m/s - the car comes to a stop), we can calculate the 
distance traveled by the car as follows:

   vi = 25 m/s                 (initial velocity)
   vf = 0 m/s                  (final velocity)
    a = - 4.73 m/s^2           (acceleration)

               vf^2 = vi^2 +2*a*d

Can you take it from here?  I hope this helped!

-Doctor Luis,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/