We were discussing the importance of friction
i.e.Â positive
and negative effects of friction,Â classifications
of friction,Â coulomb'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.Â

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####
V

####
âˆ‘F = m x a

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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.Â Â

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**The minimum stopping distance for a car or a vehicleÂ **

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.Â

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.Â Â

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

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

Initial velocity of the vehicle, V

_{0}= vÂ
Final velocity of the vehicle, V

_{f}= 0Â
Acceleration of the vehicle = - aÂ

We are taking negative sign here as there will be retardation
or deceleration after application of brake to vehicle.Â

Minimum stopping distance of the vehicle = dÂ

Mass of the vehicle = mÂ

Static coefficient of friction between the tyre and
the road = Âµ

_{sÂ }
Maximum frictional force generated due to braking
action between the tyre and road = f

_{maxÂ Â }_{}

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.Â

Normal reaction = N = mgÂ

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

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V_{f}^{2} = V_{0}^{2 }+
2 a d Â 0 = v^{2} + 2 a d

d = - v^{2} / 2 a

The minimum stopping distance for a vehicle, d = - v

^{2}/ 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 Â

####
âˆ‘F = m x a

- f_{max} = m x a

- Âµ_{s} N = m x a

- Âµ_{s} mg = m x a

a = - Âµ_{s} g

The minimum stopping distance for a vehicle, d = - v

^{2}/ 2 (- Âµ_{s}g)
The minimum stopping distance for a vehicle, d = v

^{2}/ 2 Âµ_{s}gÂ Â
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.Â

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

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

DoÂ you have any suggestions? Please write in comment box and also drop your email id in the given mail box which is given at right hand side of page for further and continuous update fromÂ www.hkdivedi.com.Â Â

### Reference:Â Â

Engineering Mechanics, By Prof K. RameshÂ Â

Image courtesy: GoogleÂ Â Â

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