THE ROLE OF MASS

Our three examples, projectile motion, motion in a circle, and projectile motion with air resistance, all demonstrate that a force produces an acceleration in the direction of the force. The next question is – how much acceleration? Clearly not all forces have the same effect. If I shove a child’s toy wagon, the wagon might accelerate rapidly and go flying off. The same shove applied to a Buick automobile will not do very much.
There is clearly a difference between a toy wagon and a Buick. The Buick has much more mass than the wagon, and is much less responsive to my shove. In our recoil definition of mass and illustrated in Figure (1), we defined the ratio of two masses as the inverse ratio of their recoil speeds

The intuitive idea is that the more massive the object, the slower it recoils. The more mass, the less responsive it is to the shove that pushed the carts apart. Think about the spring that pushes the cart apart in our recoil experiment. Once we burn the thread holding the carts together, the spring pushes out on both carts, causing them to accelerate outward. If the spring is pushing equally hard on both carts (later we will see that it must), then we see that the resulting acceleration and final velocities are inversely proportional to the mass of the cart. If m₁ is twice as massive as m₂, it gets only half as much acceleration from the same spring force. Our recoil definition and experiments on mass suggests that the effectiveness of a force in producing an acceleration is inversely proportional to the object’s mass. For a given force, if you double the mass, you get only half the acceleration. That is the simplest relationship between force and mass that is consistent with our general experience, and it turns out to be the correct one.

Figure 1: Definition of mass. When two carts recoil from rest, the more massive cart recoils more slowly.