Newton’s Laws of Motion: Understanding Force and Acceleration
Newton’s Second Law
Newton’s Second Law states that when a force is applied to an object (a “body”), it accelerates in the direction of the force. The acceleration is directly proportional to the magnitude of the force and inversely proportional to the mass of the object:
a = F / m or F = ma
The Second Law provides an explicit formula and is therefore one of the most useful. However, it can also be one of the most confusing for physics students. The problem arises when expressed as follows:
A formula in which all quantities are defined can be used to infer one of the others. A formula where one quantity is not defined may, in the best cases, serve as a definition: isolate that quantity on one side of the equal sign and define it in terms of the others. A formula in which two quantities are undefined, say A and B, is less than worthless. It tells us nothing about those numbers, since any value chosen for A or B can always be adjusted so that the equation is fulfilled. That seems to be the case in F = ma.
Acceleration is a well-defined quantity, the change in speed per second (and also has a direction). But what about m and F? How do you use the equation if it does not define them independently?
Good question. Generations of physics students, and their teachers, have often struggled with this. Some have tried defining mass and weight, using gravity as a tool. A distinguished professor of physics once swung his arm in front of the class and defined force by analogy to that produced by your muscles.
Newton’s Third Law
Newton’s Third Law, the law of reaction, states that forces never occur individually, but in equal and opposite pairs. Whenever a gun fires a bullet, it gives a kick. Firefighters pointing a hose nozzle at a fire must hold it securely, because when the water flows out, the hose recoils strongly (garden sprinklers operate on the same principle). Similarly, the forward motion of a rocket is due to the reaction pressure of the jet of hot gas coming out of its back.
Those familiar with small boats know that before jumping from the boat to land, it is wiser to tie the boat before jumping. If not, the boat “magically” moves away from the dock, making it very likely you’ll miss your jump and push the boat out of reach. This is explained by Newton’s Third Law: when your legs push your body toward the dock, the boat also applies an equal and opposite force, pushing it away from the dock.
Mach’s Formulation of Newton’s Laws
Ernest Mach, who lived in Germany two centuries after Newton, provided what may be the most satisfactory answer. Mach argued that Newton’s laws could be united into one:
“When two compact objects (“point masses” in the words of physics) act on each other, they accelerate in opposite directions, and the ratio of their accelerations is always the same.”
No mention of forces or masses, only acceleration, which can be measured. When one considers a shotgun firing, a rocket on its jet, or the Sun on the Earth (on a scale of distance that separates the Sun and Earth, they can be viewed as compact objects), the accelerations are always oppositely directed.