## How does kinetic energy affect the stopping distance of a small vehicle compared to a large vehicle – quora gas knife

The stopping distance depends on the amount of deceleration or negative acceleration that can be achieved. Now suppose both the light and heavy vehicle have the same speed but the heavy vehicle is a factor H times heavier. Then its kinetic energy is (1/2×mv^2) also H times more.

To stop the vehicle its speed v needs to be reduced to 0. The rate of reduction per unit time is dv/dt = F/m. Here F is a negative value because it’s the force that the friction of the road exerts on the tyres of the vehicle. So to achieve the same deceleration or stopping distance for the heavy vehicle as for the light vehicle we need more negative F for the heavier vehicle. Assuming the same set of tyres this is usually somewhat true. Because gravity pulls the heavier vehicle harder on the road the frictional force will be larger. But experience teaches that the amount of friction does not increase exactly proportional to the mass but somewhat less. Conclusion is that for larger m the negative force F will not grow proportionally thus leading to a deceleration that is less in magnitude. In other words the heavier vehicle will have a larger stopping distance.

As far as the relation of stopping distance to kinetic energy goes, that seems to be an indirect relation. Kinetic energy increases with m and with v, but kinetic energy alone doesn’t tell you directly what these values are. So you could have the same kinetic energy, with different m and v, leading to different stopping distances because m and v play different roles as described above.

Statistically however, larger kinetic energy correlates with higher m and v, thus showing larger stopping distances. It is probably this statistical relationship that creates the impression of a direct cause and effect relation even though it is not so.

To stop a vehicle large or small, you need to absorb or remove its kinetic energy. This is done in the brakes, where the kinetic energy is changed to heat in normal braking or changed to electric power and used to charge a battery in more modern magnetic braking systems. The brake force needed to oppose the vehicle is action, the reaction comes from the ground.. that is the friction between the rubber tyres and the tarmac. Considering a simplified case for a vehicle of mass m moving at velocity v;

KE=.5 m v^2; The friction force f is proportional to the weight of the vehicle F=k m g; the work done of a distance L is given by ; W=F L=k m g L, where k is a coefficient of friction between the tyre and road. Equating KE and W gives; L=.5v^2/k g. Hence the stopping distance depends on the velocity square and inversely proportional to the coefficient of friction. That is why the safe separation distance needed between vehicles on the road increases rapidly as the velocity increases because of the square on v (doubling the velocity increases the braking distance four folds). This is also why you need to leave more distance when the road is wet- because the coefficient of friction is smaller due to the water acting as a lubricant. Also that is why old bold tyres are less effective in braking due to the reduction in k. The shape of the treed on tyres is a result of research on how to increase the grip with the road on dry as well as wet conditions.

The stopping distance does not depend on the mass of the vehicle as seen above. But for this to be true, the tyres must be adequately strong.. That is why some ‘remold’ tyres start peeling-off on hard braking. The friction we are talking about is rolling friction which is larger than sliding friction. This explains why we use ABS braking systems in order to reduce braking force (!) whenever the sensor finds that the wheels are sliding and not rotating.. after rotation starts again, the braking goes back to normal and so on.. when you look on the tarmac after a hard braking by a lorry you see discrete and not continuous braking marks- a result of this lock and unlock of the wheels affected by the ABS braking system. The shape and length of the braking marks on the road are a good indicator of the velocity of the vehicle when it started braking, and is frequently used in vehicle accident investigations.