4.24: Calculating Acceleration from Force and Mass
Xander goes airborne on his scooter as he exits a halfpipe at Newton’s Skate Park. How did he gain enough speed in the halfpipe to fly into the air when he got to the top? His increase in speed was due partly to the force of gravity.
Acceleration, Force, and Mass
A change in an object’s motion—such as Xander speeding up on his scooter—is called acceleration. Acceleration occurs whenever an object is acted upon by an unbalanced force. The greater the net force acting on the object, the greater its acceleration will be, but the mass of the object also affects its acceleration. The smaller its mass is, the greater its acceleration for a given amount of force. Newton’s second law of motion summarizes these relationships. According to this law, the acceleration of an object equals the net force acting on it divided by its mass. This can be represented by the equation:
\begin{align*}\mathrm{Acceleration=\frac{Net\;force}{Mass}}\end{align*}
or \begin{align*}\mathrm{a=\frac{F}{m}}\end{align*}
Calculating Acceleration
This equation for acceleration can be used to calculate the acceleration of an object that is acted on by a net force. For example, Xander and his scooter have a total mass of 50 kilograms. Assume that the net force acting on Xander and the scooter is 25 Newtons. What is his acceleration? Substitute the relevant values into the equation for acceleration:
\begin{align*}\mathrm{a=\frac{F}{m}=\frac{25N}{50kg}=\frac{0.5N}{kg}}\end{align*}
The Newton is the SI unit for force. It is defined as the force needed to cause a 1kilogram mass to accelerate at 1 m/s^{2}. Therefore, force can also be expressed in the unit kg ∙ m/s^{2}. This way of expressing force can be substituted for Newtons in Xander’s acceleration so the answer is expressed in the SI unit for acceleration, which is m/s^{2}:
\begin{align*}\mathrm{a=\frac{0.5N}{kg}=\frac{0.5 kg*m/s^2}{kg}=0.5 m/s^2}\end{align*}
Q: Why are there no kilograms in the final answer to this problem?
A: The kilogram units in the numerator and denominator of the fraction cancel out. As a result, the answer is expressed in the correct SI units for acceleration.
Calculating Force
It’s often easier to measure the mass and acceleration of an object than the net force acting on it. Mass can be measured with a balance, and average acceleration can be calculated from velocity and time. However, net force may be a combination of many unseen forces, such as gravity, friction with surfaces, and air resistance. Therefore, it may be more useful to know how to calculate the net force acting on an object from its mass and acceleration. The equation for acceleration above can be rewritten to solve for net force as:
Net Force = Mass x Acceleration, or
F = m x a
Look at Xander in the picture below. He’s riding his scooter down a ramp. Assume that his acceleration is 0.8 m/s2. How much force does it take for him to accelerate at this rate? Substitute the relevant values into the equation for force to find the answer:
F = m x a = 50 kg x 0.8 m/s^{2} = 40 kg ∙ m/s^{2}, or 40 N
Q: If Xander and his scooter actually had a mass of 40 kg instead of 50 kg, how much force would it take for him to accelerate at 0.8 m/s^{2}?
A: It would take only 32 N of force (40 kg x 0.8 m/s^{2}).
At the URL below, you can review how to calculate net force from mass and acceleration and try some practice problems. If you want to try more challenging problems, click on the relevant links at the bottom of the Web page.
http://www2.franciscan.edu/academic/mathsci/mathscienceintegation/MathScienceIntegation856.htm
Summary
 According to Newton’s second law of motion, the acceleration of an object equals the net force acting on it divided by its mass, or \begin{align*}\mathrm{a=\frac{F}{m}}\end{align*}
a=Fm .  This equation for acceleration can be used to calculate the acceleration of an object when its mass and the net force acting on it are known.
 The equation for acceleration can be rewritten as F = m x a to calculate the net force acting on an object when its mass and acceleration are known.
Vocabulary
 acceleration: Measure of the change in velocity of a moving object.
Practice
At the following URL, solve the six problems on page 2.
http://www2.fcsconline.org/staff/ferrisa/Kinesiology/Force%20WS.pdf
Review
 What is the equation for calculating the acceleration of an object when its mass and the net force acting on it are known?
 Xander’s friend Corey has a skateboard that he rides at Newton’s Skate Park. That’s Corey doing a jump in the picture below. The combined mass of Corey and his skateboard is 60 kg. At the top of his jump, the net force acting on him is 30 Newtons. What is his acceleration at that moment?

 What net force would have to act on Cory for him to have an acceleration of 1 m/s^{2}?
Image Attributions
 Describe the relationships among acceleration, force, and mass.
 Show how to calculate acceleration from force and mass.
 Show how to calculate force from mass and acceleration.