Re: if a truck's mass is doubled, is it's stopping distance unchanged upon braking?

this question is about a truck that slams on it's brakes and stops. If the mass is doubled, the stopping distance is aparently the same. My physics class cannot understand why this is so, and the teacher can't even answer it. According to the teacher's test key, this is so, but no one knows why.

Hello,
This is an interesting question. Intuition, and even my initial thoughts were that the heavier truck would take a longer distance to stop. However, the correct answer, theoretically, is that both trucks will take the same distance to stop.

The stopping distance is directly related to the total friction force developed between the tires and the road surface. This friction force, applied over the stopping distance must dissipate the kinetic energy of the truck.

In your physics class you probably studied the friction formula:
$F\; =\; u\; N$ (formula 1)
where 'F' is the horizontal friction force, 'u' is the coefficient of friction between the tires and the road, and 'N' is the normal force, equal to the weight of the truck. ('u' is usually shown with the Greek letter 'mu', but I don't have that access to that symbol here).

We can rewrite 'N', the weight of the truck, as 'mg', where m is the mass of the truck and g is gravity.

The friction formula becomes: $F\; =\; u\; (mg)$ (formula 2)

Let's jump for a minute to the other concept- kinetic energy. On level ground, it is the kinetic energy of the vehicle that must be dissipated, or converted to another form of energy (heat) to bring that vehicle to a stop. $Kinetic\; Energy\; (KE)\; =\; 1/2\; m\; v2$

I will skip some of the math that connects the two formulas, but you can see that the friction force from formula 2 is the stopping force to remove the kinetic energy of the truck.

NOW, notice that the mass 'm' appears in both formulas. This means that if the mass of the truck doubles, the kinetic energy will double, but that the friction stopping force will also double. In other words, the stopping distance is independent of the mass (or weight) of the truck.

This answer is theoretical. In the real world there would be two conditions for the stopping distance of the two trucks to be the same: 1) All of the truck brakes have sufficient 'power' to stop (lock up) each wheel when applied. 2) Every wheel in contact with the ground has this braking capacity. Remember, the shortest stopping distance for either truck is when the brakes are applied hard enough so that all tires are 'just ready' to skid on the pavement.

I hope that this helps!

Best Regards,

Jay Shapiro