## Math in the Physical Sciences## Classical Physics and Mathematics |

## How is mathematics used in physics to describe motion? |

Everything in the universe is in motion, from the rotating Earth to subatomic particles. Motion in physics is described mainly through mathematics, including speed, velocity, acceleration, momentum, force (something that changes the state of rest or motion of an object), torque (when a force causes rotation or twisting around a pivot point), and inertia (a body at rest remains at rest, and a body in motion remains in motion, until acted upon by an outside force).

Unlike what most people think, speed and velocity are not the same. Speed is the rate at which something moves; velocity is speed in a certain direction. Speed is also called a scalar quantity, described by the following formula: speed = distance/time. For example, if you drive 200 miles in 2 hours, and your speed is constant, your average speed is 200/2, or 100 miles per hour. On the other hand, velocity is known as a vector quantity (for more about vectors, see “Mathematical Analysis”). That gives velocity both speed and direction—and that leads directly to acceleration.

When an object’s velocity changes, we say that it accelerates. Acceleration—also a vector like velocity—is represented as the change in velocity divided by the time it takes for the change to occur. We define the formula for acceleration—or the change in velocity per unit time—as *a* = Δ*v*/Δ*t*. In this equation, *a* is acceleration, Δ*v* is change in velocity of an object (the delta symbol stands for change), and Δ*t* is the change in time needed to reach the velocity. For example, if acceleration is constant, and a person drove from a standing point to 60 miles per hour in 5 seconds, the equation becomes: 60 miles per hour / 5 seconds (or final speed minus the initial speed, all divided by the elapsed time). This means that the acceleration is equal to 17.6 feet per second squared (you have to change the miles per hours to feet and seconds, respectively).

Momentum relates to the amount of energy maintained by a moving object; it is also defined as the force necessary to stop an object from moving. It depends on the mass and velocity of an object, and is represented as: *M = mv*, in which *M* is momentum, *m* is the mass of the object, and *v* is the object’s velocity.