Car Mass & Speed: Explaining with Newton's 1st Law

[cremedelacreme]

If a car of mass 50 kg and another one of mass 100 kg rolls down a slide, which one will move faster? Why? Can it be explained using Newton's 1st Law: Law of Inertia? or the Kinetic Energy?

 

[malawi_glenn]

If we assumume no air resistance, and we let two bodies fall free from the same height; body#1's mass is 3000kg, and body#2's mass is 1kg, which of these two will hit the ground first?

Same problem.

 

[Ks. Jan Jenkins]

Inertia states that a body has the curious property that it naturally stays at the velocity at which it is - velocity is somehow a property of the body, such that it requires something from outside to change it. Thus it naturally stays at rest (velocity = 0) or is moving in a certain direction and certain speed (as velocity is a vector). This is Newton's first law, which simply states that there is something we call 'momentum' which is the product of an objects mass by its velocity. Momentum thus neatly gives three fundamental qualities (mass, length, time) of a physical object - its mass (kg) times its velocity (m / s).

Newton's second law simply states how this momentum can change. In order to change momentum, something must act upon the object. This acting upon an object, changing one of these fundamental qualities (mass, length, time) we call a force, which is the change of momentum over a certain time (F=∂p/∂t). In many simple problems the mass is not changed, so we can write this as the mass times the change of velocity over time or acceleration (F=m*a). A direct consequence is that if there is no change of momentum, there can be no net force acting on the object (F=∂p/∂t=0).

Newton's third law concerns the nature of an interaction - what happens when one object (which has a certain momentum) interacts (hits or collides) with another object (which has its own momentum)? From the second law we know that the change of momentum of the first must be the force exerted by the other. What Newton's third law adds is that the entire momentum of the entire system is conserved - which in a way follows from the first law saying that the object keeps its qualities unless something outside acts upon it. Thus taking the two objects together, we know that the momentum beforehand of both of the objects will be the same as the momentum after their interaction. As a consequence the forces that the two objects are subjected to must be equal and opposite. This is often written as the 'reaction' of the one object upon the other 'equal and opposite', though the conservation of momentum is the more fundamental idea.

There are also other profound consequences that can be derived from these three laws - that of the existence of a center of mass for instance and also something we call 'angular momentum' of an object that is spinning about a certain axis.

The question of Kinetic Energy is related, but is actually a wholely different property. It is what follows when we change the momentum of a body through a certain distance - or as it is more commonly stated, the exerting of a force through a certain distance.

As this distance is the product of velocity times the time taken through this distance (v = d/∂t -> d = v*∂t), and the force a change of momentum over time (∂p/∂t)we see that it is the change of the product of the velocity times the momentum ∂(v.p) = ∂(v.mv). Assuming constant mass (and using the product rule from calculus) we can show that this work done is equal to one-half of the square of the change of velocity times the mass.

One the other hand we call energy the 'capacity to do work' - it is not the performing of the work, but the capacity to do it. Thus kinetic energy is exactly that capacity to effect that work (1/2 m v^2).

Concerning your problem, let us look at it using these laws and then the concept of energy :

1st and 2nd law : the two cars would normally stay at their current velocity unless something is acting upon them - what is acting upon them ? Well in this case there is gravity, which is pulling them to the center of the Earth - so each of them experiences this force in proportional to their mass (the acceleration due to gravity = 9.81 m/s^2). Thus the Earth exerts a force (F=m*a) on the larger car of 100 kg * 9.81 m/s^2 = 981 kg*m/s^2. The force on the smaller car is 49.05 kg*m/s^2. However, and what is more interesting, is that these same objects want to resist this change of momentum (inertia - 1st law) which is exactly in proportion to their mass. As a consequence they will both change their velocity at the same rate - they will both fall at the same rate or 9.81 m/s^2. There is a small catch however, which I will explain at the end.

However, looking at their respective kinetic energies, there is an enormous difference! Why? because the kinetic energy is that capacity to do work over a certain distance, or as we have seen, the change of momentum times the velocity. Since the change of momentum of a larger object is greater, the resulting kinetic energy will be greater. Thus the larger car, will have twice as much kinetic energy at the same velocity than the smaller one.

There is a small problem with your problem though! You state that the cars are ROLLING down a slide - thus there is not only a question of the change of momentum, but also the change of the moment of inertia (the tires are spinning !). The moment of inertia is the mass times the radius of the object spinning squared. Say the tires are the same on each car - yet is the force exerted on the tires (or rather, torque) the same ?

For the final answer, I suggest you do an experiment! Take one of your younger brothers toys, a dump truck, and put it on an inclined piece of wood. Time how long it takes to reach the bottom. Then put some weight in it. Did it take the same amount of time or less time than before ? Notice that you only changed one thing - the mass. What could make the difference if there is any? Maybe make a race with two identical trucks (one empty and one full) and find out for yourself !

Hope this was helpful, even if a little long-winded.

Ks. Jan Jenkins +
(physics professor on vacation, who can't wait to get back to class!)