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 laws. 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 body's mass, you can immediately calculate the acceleration it experiences.

## The Average Acceleration Equation

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, *x*_{1} and *x*_{2,} 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 *x*_{1} is *t*_{1,} and the time at *x*_{2} is *t*_{2}, the car's speed (*s*) is:

*s*= ∆*x* ÷ ∆*t* = (*x*_{2} − *x*_{1}) ÷ (*t*_{2} − *t*_{1}).

Now suppose the car's speedometer is working, and it records two different speeds at points *x*_{1} and *x*_{2}. 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 be a negative number, which would mean that the car was decelerating. If the instantaneous speed as recorded by the speedometer at time *t*_{1} is *s*_{1}, and the speed at time *t*_{2} is *s*_{2}, the acceleration (*a*) between points *x*_{1} and *x*_{2} is:

*a* = ∆*s* ÷ ∆*t* = (*s*_{2} − *s*_{1}) ÷ (*t*_{2} − *t*_{1}).

This average acceleration equations 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/s^{2}. In the imperial system, the preferred units of acceleration are feet/second/second, or ft/s^{2}.

**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 400 mph, and the time is 30 minutes, which is 0.5 hours. The acceleration is then

*a* = 400 mph/0.5 hr = 800 miles/hr^{2}.

## Newton's Second Law Provides an Acceleration Calculator

The equation that expresses Newton's second law, *F* = *ma*, is one of the most useful in physics and serves as an acceleration formula. 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/s^{2}. 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).

You can rearrange Newton's equation to solve for acceleration by dividing both sides by *m*. You get:

*a* = *F*/*m*

Use this expression as an acceleration calculator when you know the mass and the magnitude of the applied force.

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

*a* = *F*/*m* = 20 N/8 kg = 2.5 m/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/s^{2}. The answer is 62.5 slugs (slugs are the unit for mass in the imperial system). Now you can calculate acceleration:

*a* = *F*/*m* = 1,000 lbf/62.5 slugs = 16 ft/s^{2}.

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