Understanding Speed and Velocity

Fundamental Quantities Crucial to Understanding Physics of Motion

Jan 5, 2008

Speed and velocity are fundamental quantities needed to understand the physics of motion. They have precise meanings different from everyday use.

Physics of Motion

Mechanics is the branch of physics that deals with motion. Velocity, speed, and, acceleration are among the most fundamental quantities in mechanics. Like many quantities in physics, they have very precise meanings that are different from their less precise everyday meanings. Physics students need to clearly understand the precise physical meanings of these quantities.

Difference Between Speed and Velocity

Most people consider speed and velocity as interchangeable terms. In physics they are not. They have similar but distinct meanings.

The difference is direction. Velocity includes direction. Speed does not. A car might travel with a speed of 50 miles per hour and a velocity of 50 miles per hour towards the east. When physicists use the word velocity, they understand that the direction is included. When they say speed, they understand that the direction is unspecified.

A quantity that includes direction is a vector quantity, and a quantity that does not is a scalar. Velocity is a vector; speed is a scalar. Be aware of this distinction when reading or talking physics.

Average Velocity

Velocity is our way of measuring the rate of change in an object's position. Its units are a distance unit divided by a time unit, such as meters per second or miles per hour.

If you drive 50 miles in your car and it takes an hour, your average speed is 50 miles per hour. The average velocity or speed is defined as the change in position divided by the change in time. The change in position is the distance covered, and the change in time is the time it takes. In equation form it is:

(average velocity) = (change in position)/(change in time)

Instantaneous Velocity

Few people while driving that 50 miles do so at a constant speed. They slow down in the towns and speed up on the highway. So physicists also define the instantaneous velocity.

One might drive on the highway at 65 miles per hour. The average velocity over just the portion of the trip on the highway will be closer to the true instantaneous velocity on this part of the trip. In general the smaller the time interval used for finding the average velocity, the closer the average velocity is to the true instantaneous velocity. Ideally the instantaneous velocity is the average velocity over a time interval of 0 seconds. However that would require dividing by zero in the equation above.

Physicists therefore find the instantaneous velocity by looking at the average velocity over successively smaller and smaller time intervals. As the time interval approaches zero, the average velocity approaches the instantaneous velocity. This process is the mathematical process of taking the limit.

The instantaneous velocity is therefore the limit as the change in time approaches 0 of the change in position divided by the change in time. For those familiar with calculus, this limit is the definition of the derivative. The instantaneous velocity is therefore the derivative of the position with respect to time, or v=dx/dt.

Acceleration

Just as velocity measures the rate of change of an object's position, the acceleration measures the rate of change of the object's velocity. If an object's velocity is changing in any way, the object is accelerating. Many people think of accelerating as speeding up, but accelerating includes speeding up, slowing down, or changing direction.

Understanding speed, velocity, and acceleration is crucial to understanding the physics of motion.