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HOW TO CALCULATE THE MINIMUM STOPPING DISTANCE OF A CAR

We were discussing the importance of friction i.e. positive and negative effects of frictionclassifications of frictioncoulomb's law of dry friction, some guidelines for solving frictional problems, concept of rolling resistance or rolling friction and wedge friction and concept of self- locking in engineering mechanics with the help of our previous posts. 

Now, we will be interested further to understand here a very important concept in engineering mechanics i.e. the minimum stopping distance for a vehicle with the help of this post.  

The minimum stopping distance for a car or a vehicle 

Let us consider that a vehicle of mass m is moving with a speed v along a level road as displayed here in following figure. 

As car or any vehicle is initially running with a speed of v. Now, we are applying the brake in order to stop the vehicle i.e. car.  

We want to determine here the minimum stopping distance d for a vehicle of mass m moving with speed v along a level road.  


Let us assume and write down here the following terms from above figure. 

Initial velocity of the vehicle, V0 = v 
Final velocity of the vehicle, Vf = 0 

Acceleration of the vehicle = - a 
We are taking negative sign here as there will be retardation or deceleration after application of brake to vehicle. 

Minimum stopping distance of the vehicle = d 
Mass of the vehicle = m 
Static coefficient of friction between the tyre and the road = µ
Maximum frictional force generated due to braking action between the tyre and road = fmax  

Vehicle will be stopped with minimum stopping distance when retardation will be maximum and it is only possible when frictional force generated will be maximum. Hence, we are considering here the maximum frictional force. 

Normal reaction = N = mg 

Let us write and use the following equation of motion as mentioned below

Vf2 = V02 + 2 a d  0 = v2 + 2 a d
d = - v2 / 2 a

The minimum stopping distance for a vehicle, d = - v2 / 2 a

Let us draw here the free body diagram of the vehicle and write here the equation of motion of the vehicle as mentioned below  

∑F = m x a
- fmax = m x a
- µs N = m x a
- µs mg = m x a
a = - µs g

The minimum stopping distance for a vehicle, d = - v2 / 2 (- µs g)
The minimum stopping distance for a vehicle, d = v2 / 2 µs g  


Therefore, we can see here that the minimum stopping distance for a vehicle will be dependent over the velocity of the vehicle and co-efficient of friction between the tyre and road. 

Usually, the static co-efficient of friction between the tyre and road is 0.8.  


Therefore, we have seen here the minimum stopping distance for a vehicle with the help of this post. 

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We will find out now the concept of instantaneous center of zero velocity in our next post.   

Reference:  

Engineering Mechanics, By Prof K. Ramesh  
Image courtesy: Google     

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