Problems involving calculating speed, velocity and acceleration commonly appear in physics. Often these problems require calculating the relative motions of trains, planes and automobiles. These equations can also be applied to more complex problems like the speeds of sound and light, the velocity of planetary objects and the acceleration of rockets.

#### TL;DR (Too Long; Didn't Read)

Equations for speed, velocity and acceleration depend on change of position over time. Average speed uses the equation "speed equals distance traveled (d) divided by travel time (t)," or average speed = d ÷ t. Average velocity equals speed in a direction. Average acceleration (a) equals change in velocity (Δv) divided by the time interval of the velocity change (Δt), or a = Δv ÷ Δt.

## Formula for Speed

Speed refers to distance traveled during a period of time. The commonly used formula for speed calculates average speed rather than instantaneous speed. The average speed calculation shows the average speed of the entire journey, but instantaneous speed shows the speed at any given moment of the journey. A vehicle's speedometer shows instantaneous speed.

Average speed can be found using the total distance travelled, usually abbreviated as d, divided by the total time required to travel that distance, usually abbreviated as t. So, if a car takes 3 hours to travel a total distance of 150 miles, the average speed equals 150 miles divided by 3 hours, equals an average speed of 50 miles per hour (150 ÷ 3 = 50).

Instantaneous speed actually is a velocity calculation that will be discussed in the velocity section.

Units of speed show length or distance over time. Miles per hour (mi/hr or mph), kilometers per hour (km/hr or kph), feet per second (ft/s or ft/sec) and meters per second (m/s) all indicate speed.

## Formula for Velocity

Velocity is a vector value, meaning that velocity includes direction. Velocity equals distance traveled divided by time of travel (the speed) plus the direction of travel. For example, the velocity of a train traveling 1,500 kilometers eastward from San Francisco in 12 hours would be 1,500 km divided by 12 hr east, or 125 kph east.

Going back to the problem of the car's speed, consider two cars starting from the same point and traveling at the same average speed of 50 miles per hour. If one car travels north and the other car travels west, the cars do not end up in the same place. The velocity of the northbound car would be 50 mph north, and the velocity of the westbound car would be 50 mph west. Their velocities are different even though their speeds are the same.

Instantaneous velocity, to be completely accurate, requires calculus to evaluate because to approach "instantaneous" requires reducing the time to zero. An approximation can be made, however, using the equation instantaneous velocity (v_{i}) equals change in distance (Δd) divided by change in time (Δt), or v_{i} = Δd ÷ Δt. By setting the change of time as a very short period of time, a nearly instantaneous velocity can be calculated. The Greek symbol for delta, a triangle (Δ), means change.

For example, if a moving train has travelled 55 km east at 5:00 and reached 65 km east at 6:00, the change in distance is 10 km east with a change of time as 1 hour. Inserting these values into the formula v_{i} = Δv ÷ Δt gives v_{i} =10 ÷ 1, or 10 kph east (admittedly a slow velocity for a train). The instantaneous velocity would be 10 kph east, read on the engine's speedometer as 10 kph. Of course, an hour isn't "instantaneous," but it serves for an example.

Suppose instead that a scientist measures the change of position (Δd) of an object as 8 meters over a time interval (Δt) of 2 seconds. Using the formula, the instantaneous velocity equals 4 meters per second (m/s) based on the calculation v_{i} = Δd ÷ Δt, or v_{i} = 8 ÷ 2 = 4.

As a vector quantity, instantaneous velocity should include a direction. Many problems, however, assume that the object continues traveling in the same direction during that short interval of time. The directionality of the object is then ignored, which explains why this value is often called instantaneous speed.

## Equation for Acceleration

What's the formula for acceleration? Research shows two apparently different equations. One formula, from Newton's second law, relates force, mass and acceleration in the equation force (F) equals mass (m) times acceleration (a), written as F = ma. Another formula, acceleration (a) equals change in velocity (Δv) divided by change in time (Δt), calculates the rate of change in velocity over time. This formula may be written a = Δv ÷ Δt. Since velocity includes both speed and direction, changes in acceleration may result from changes in speed or direction or both. In science, the units for acceleration usually will be meters per second per second (m/s/s) or meters per second squared (m/s^{2}).

These two equations, F = ma and a = Δv ÷ Δt, are not at odds with each other. The first shows the relationship of force, mass and acceleration. The second calculates acceleration based on change in velocity over a period of time.

Scientists and engineers refer to increasing velocity as positive acceleration and decreasing velocity as negative acceleration. Most people, however, use the term deceleration instead of negative acceleration.

**Acceleration of Gravity**

Near the Earth's surface, the acceleration of gravity is a constant: a = -9.8 m/s^{2} (meters per second per second or meters per second squared). As Galileo suggested, objects with different masses experience the same acceleration from gravity and will fall at the same speed.

## Online Calculators

By entering data into an online speed calculator, acceleration can be calculated. Online calculators can be used to compute the equation of speed to acceleration and force. Using an acceleration and distance calculator requires knowing speed and time as well.

#### Warnings

Using an online calculator to complete homework might not be acceptable to the teacher. However, using them to double-check your homework might be considered an ethical use of these calculators. Check with the teacher.

#### References

- Oregon State University: Average Speed
- Georgia State University HyperPhysics: Description of Motion
- Rochester Institute of Technology: Displacement, Velocity, Acceleration
- University of Rochester: Newton's Three Laws of Motion
- University of California, Santa Cruz: Instantaneous Velocity
- University of Florida: Physics Intro & Kinematics
- Loyola University Chicago: Deriving Relationships Among Distance, Speed and Acceleration
- Road and Track: Anatomy of a High-Speed Car Crash
- Georgia State University HyperPhysics: Force on Driver in Example Car Crash
- Stanford Solar Center: Galileo
- Omni Calculator: Acceleration Calculator