How To Calculate Acceleration

Speed and acceleration are two fundamental concepts in mechanics, or the physics of motion, and they are related. If you measure the speed of an object while you record the time, then measure it again a little later, also while recording the time, you can find acceleration, which is the difference in those speeds divided by the time interval. That's the basic idea, although in some problems, you might have to derive speeds from other data.

There's another way to calculate acceleration based on Newton's second law of motion. According to the first law, a body remains in a state of uniform motion unless acted upon by a force, and the second law expresses the mathematical relationship between the magnitude of the force (‌F‌) and the acceleration (‌a‌) a body of mass ‌m‌ experiences because of that force. The relationship is ‌F = ma‌. If you know the magnitude of a force acting on a body, and you know the mass of the object, you can immediately calculate the acceleration it experiences.

\(\text{Average Acceleration} = \frac{\text{Change in Speed}}{\text{Time Interval}} = \frac{{\Delta}v}{{\Delta}t}\)

Average Velocity of an Object

Think of a car on a highway. If you want to know how fast it's going, and the speedometer isn't working, you pick two points on its path, x1 and x₂, and you look at your clock as the car passes each point. The car's average speed is the distance between the two points divided by the time it takes for the car to pass both of them. If the time on the clock at x₁ is t₁ and the time at x₂ is t₂, the car's speed (s) is:

\(\text{Speed}= s =\frac{\Delta x}{\Delta t}=\frac{x_2-x_1}{t_2-t_1}\)

An important part of measuring and describing speed is the direction of motion. After calculating speed, which is called a scalar quantity, we then ask where the object was traveling with that speed: north, south, up, away, etc. When we add direction to the quantity of speed it becomes velocity. Velocity is called a vector quantity because it includes this bonus information about direction, and it is extremely useful in high school math and science and beyond.

Now suppose the car's speedometer is working, and it records two different speeds at points x₁ and x₂. Since the speeds are different, the car had to be accelerating. Acceleration is defined as the change of speed over a particular time interval. It can also be a positive or negative number. When acceleration is positive, the car is accelerating to higher speeds; if there is negative acceleration, the car is decelerating to lower speeds. If acceleration is zero, then the object has a constant velocity.

If the instantaneous speed (also called initial velocity) as recorded by the speedometer at time t₁ is s₁, and the speed at time t₂ is s₂ (final velocity), the acceleration (a) between points x₁ and x₂ is:

\(\text{Acceleration} = a =\frac{\Delta s}{\Delta t}=\frac{s_2-s_1}{t_2-t_1}\)

This average acceleration equation tells you that if you measure the speed at a certain time and measure it again at another time, the acceleration is the change of speed divided by the time interval. The units of speed in the SI system are meters/second (m/s), and the units of acceleration are meters/second/second (m/s/s) which is usually written m/s². In the imperial system, the preferred units of acceleration are feet/second/second, or ft/s².

‌Example‌: An airplane is flying 100 miles per hour just after takeoff, and it reaches its cruising altitude 30 minutes later, when it is flying 500 miles per hour. What was its average acceleration as it climbed to its cruising altitude?

We can use the acceleration formula derived above. The difference in speed (∆‌s‌) is 500mph – 100mph to get 400 mph, and the time is 30 minutes, which is 0.5 hours. The acceleration is then

\(a=\frac{400}{0.5}=800 \frac{\text{miles}}{\text{hr}^2}\)

The equation that expresses Newton's second law, ‌F = ma‌, is one of the most useful relationships in physics. The unit of force in the SI system is the Newton (N), named after Sir Isaac himself. One Newton is the force required to give a 1 kilogram mass an acceleration of 1 m/s². In the imperial system, the unit of force is the pound. Weight is also measured in pounds, so to differentiate mass from force, units of force are called pounds-force (lbf).

To use F = ma to calculate the acceleration of the object in question, you can rearrange Newton's equation to solve for acceleration by dividing both sides by ‌m‌. You get:

\(a=\frac{F}{m}\)

Use this expression as an acceleration calculator when you know the mass and the magnitude of the applied force. It is also important to note that to find the net acceleration (total acceleration) in a situation you also need to use the net force on an object.

‌Example:‌ An object with a mass of 8 kg. experiences a force of 20 Newtons. What average acceleration does it experience?

\(a=\frac{F}{m}=\frac{20}{8}=2.5\frac{\text{m}}{\text{s}^2}\)

‌Example‌: A 2,000-pound car experiences a force of 1,000 pounds-force. What is its acceleration?

Weight is not the same as mass, so to get the car's mass, you have to divide its weight by the acceleration due to gravity, with is 32 ft/s². The answer is 62.5 slugs (slugs are the unit for mass in the imperial system). Now you can calculate acceleration:

\(a=\frac{F}{m}=\frac{1000}{62.5}=16\frac{\text{f}}{\text{s}^2}\)

Speed and Acceleration

If you also know the initial speed, you can use this calculated acceleration to determine the final speed after a given amount of time.

\(\text{Final Speed} = s_f = a {\Delta t} + s_i\)

This is actually a slightly different representation of a kinematics equation that is crucial in physics and measurements of motion.

When measuring the acceleration of an object in the previous scenarios, it was always in reference to average acceleration over a given time frame. In some cases there is constant acceleration, but oftentimes acceleration can be changing within these time intervals. When trying to describe these changes in motion we use the field of calculus and operations called derivatives and integrals.

In calculus we use the terms instantaneous acceleration and instantaneous velocity to talk about very accurate measurements of acceleration and velocity. Acceleration is still defined as the rate of change of velocity, but these measurements refer to motion at an instant instead of over a period of time. This is a very difficult concept that is also crucial to the fields of physics and calculus.