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I was wondering, how can I calculate the decelerations of an object due to friction - and therefore find the maximal distance it can reach?

For example, if a car suddenly brakes in the middle of a road, how can I find how long it will take it to fully stop, its deceleration, the distance it will reach before stopping?

Or if someone is skiing and reaches the bottom of a mountain, where the ground becomes flat, how can I know how long it will take the skier to come to a complete stop?

Before I’ve used the formula (of which I’m not too sure): $deceleration = -g \times \mu_{static}$. But now that I think of it, I’m not sure if it’s correct anymore...

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  • $\begingroup$ You can accept an answer by clicking on the tick next to the answer since you are new $\endgroup$
    – imposter
    Aug 15 '20 at 19:49
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I was wondering, how can I calculate the decelerations of an object due to friction - and therefore find the maximal distance it can reach?

If you know the velocity of the object before friction begins to bring it to a stop you can calculate the stopping distance using the work-energy theorem which states that the net work done on an object equals its change in kinetic energy. If the only force acting on the object bringing it to a stop is the friction force then

$$W_{net}=μmgd=\frac{mv^2}{2}$$

$$μgd=\frac{v^2}{2}$$

$$d=\frac{v^2}{2μg}$$

Where $d$ = stopping distance, $v$ = velocity of object before encountering friction, $μ$ = the coefficient of friction and $g$= acceleration due to gravity. In the case of vehicle braking distance, if the car is skidding you use the coefficient of kinetic friction between the tires and the road. If the wheels continue to roll you use the coefficient of static friction. Generally rolling resistance can be ignored.

You can calculate the magnitude of the deceleration from Newtons second law

$$a=\frac{F}{m}=\frac{μmg}{m}=μg$$

And finally you can calculate the stopping time from

$$d=\frac{at^2}{2}$$

$$t=\sqrt \frac{2d}{a}$$

Hope this helps.

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Well g×coefficient of static friction is an incorrect way of finding the deceleration due to friction.

The first obvious reason is that if the object is moving, then kinetic friction comes into play, or else rolling friction comes into play in real rolling conditions.

Only in a perfectly ideal pure rolling scenario can we take static friction in our calculations.

However most of the questions deal with ideal cases so this part is mostly correct.

Also the other term "g" would be correct only in cases such as a car traveling on a straight road, etc.

If the object was traveling on an incline, your formula would give you an incorrect value.

This is because of how friction is defined.

When an object is moving Friction=Coeffcient of kinetic friction × Normal force

Only on a flat surface would the normal force be mg.

On a plane inclined at an angle $\theta $ with the ground, it would be $mg \cos\theta $

So deceleration here would have a gcos $\theta $ term instead of the g term in your formula.

Hope it helped!

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    $\begingroup$ This is incorrect, friction can be static if the wheels are in pure rolling. $\endgroup$ Aug 15 '20 at 5:19
  • $\begingroup$ @Binod Thanks for pointing it out. Now I have edited it correctly. Please tell me if any other fixes are required. $\endgroup$ Aug 15 '20 at 7:44

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