Acceleration is an important concept in physics used to describe motion and solve problems. In this article, we will define acceleration, the formula for acceleration, and its units. We will review examples of how to find acceleration including positive acceleration and negative acceleration.
What We Review
Review: Kinematic Terms
There are a few kinematic terms that you’ll need to know to understand acceleration. The first kinematic term is velocity. Velocity is the rate of change of position, or the displacement, over time. Displacement is another name for the change in position. Remember that an object moving at a constant velocity has a constant change of position every second. The motion map below shows an object moving with a constant velocity.
It is also important to remember the definition of vector quantities. Vectors are quantities that have both a magnitude, or size, and direction. For example, velocity is a vector quantity because it describes both how fast an object is moving (magnitude) and the direction the object is moving in. For an in-depth review of vectors, scalars, displacement, and velocity, visit our Albert blog post introducing kinematics.
What is Acceleration?
Acceleration is defined as the rate of change of velocity. This means that if an object’s velocity is increasing or decreasing, then the object is accelerating. Unlike an object moving at a constant velocity, an accelerating object will not have a constant change in position every second. An accelerating object could cover more and more distance with every second, or less and less distance every second. The first motion map below shows an accelerating object that is speeding up and the second motion map shows an accelerating object that is slowing down.
Since acceleration depends on the change in velocity, acceleration is a vector quantity. This means that acceleration has both magnitude and direction.
The standard units of acceleration are meters per second squared, \text{m/s}^2. These units come from the units of velocity, meters per second, and the units of time, seconds. Since acceleration is the change in velocity over time, its units are the units of velocity (meters per second) divided by the units of time (second).
For more information about acceleration and some examples, watch this quick video.
Acceleration Formula
Using our definition of what acceleration is, we can put together a formula for calculating acceleration.
| Acceleration Formula a=\dfrac{\Delta v}{t} |
Here, a is acceleration, \Delta v is the change in velocity, and t is time.
Before calculating acceleration, you will often need to first calculate the change in velocity. This is the difference between the object’s final velocity, v_f, and its initial velocity, v_i.
| Change in Velocity Formula \Delta v = v_f - v_i |
How to Find Acceleration Using the Acceleration Formula
In this next section, we will go over some examples of calculating acceleration. First, we will see an example of positive acceleration, then an example of negative acceleration.
Example 1: Acceleration of a Car Speeding Up
A car starts from rest and moves forward while speeding up to 26\text{ m/s} in 8\text{ s}. Calculate the car’s acceleration.
The first step in solving this problem is to determine the change in velocity. Since the car started from rest, its initial velocity, v_i, is 0\text{ m/s}. Its final velocity is 26\text{ m/s}. Therefore, the car’s change in velocity is:
\Delta v = v_f - v_i = 26\text{ m/s} - 0\text{ m/s} = 26\text{ m/s}
Now we can calculate the car’s acceleration by dividing this change in velocity by the time of 8\text{ s}:
a=\dfrac{\Delta v}{t}
a=\dfrac{26\text{ m/s}}{8\text{ s}}
a=3.25\text{ m/s}^2
Example 2: Acceleration of a Car Slowing Down
Now let’s consider a situation where an object is slowing down.
A car initially moving forward at a velocity of 26\text{ m/s} approaches a school and slows down to a velocity of 11\text{ m/s} in 3\text{ s}. Calculate the car’s acceleration.
The first step is to find the change in velocity. The car’s initial velocity is 26\text{ m/s} and its final velocity is 11\text{ m/s}. Therefore, the car’s change in velocity is:
\Delta v = v_f - v_i = 11\text{ m/s} - 26\text{ m/s} = -15\text{ m/s}
Now we can calculate the car’s acceleration by dividing this change in velocity by the time of 3\text{ s}:
a=\dfrac{\Delta v}{t}
a=\dfrac{-15\text{ m/s}}{3\text{ s}}
a=-5\text{ m/s}^2
We will explain in the next section more about what a negative acceleration means.
Determining the Direction of Acceleration
As a vector quantity, acceleration has both magnitude and direction. The direction of acceleration depends on if the object is speeding up or slowing down, and the direction the object is moving. In general, if an object is speeding up, its acceleration will be in the same direction as its motion. If an object is slowing down, its acceleration is in the opposite direction of its motion.
Positive Acceleration
There are two types of situations where an object can have a positive acceleration. If an object is speeding up and moving in a positive direction, it has a positive acceleration. The car speeding up in the first example was an example of positive acceleration. The car is moving forward in a positive direction and speeding up, so the acceleration is in the same direction as the car’s motion.
An object can also have a positive acceleration if it is slowing down while moving in a negative direction. Since the object is slowing down, the acceleration is in the opposite direction of its motion.
Negative Acceleration
As we saw in the second example, an object can have a negative acceleration when the object is slowing down while moving in a positive direction. The car in the school zone was moving forward in a positive direction and slowing down, so the acceleration was in the opposite direction of the car’s motion.
An object can also have a negative acceleration if it is speeding up while moving in a negative direction. Since the object is speeding up, its acceleration is in the same direction as its motion.
Examples: Identify the Direction of Acceleration
Example 1: A Cyclist
Consider a cyclist riding on a straight road. The cyclist moves in a positive direction at a constant velocity. Suddenly, the cyclist sees an obstacle and applies the brakes, slowing down until they come to a complete stop.
To find the direction of acceleration for the cyclist, first, identify whether the cyclist is speeding up or slowing down. In this case, the cyclist is slowing down. Since the cyclist is slowing down, the acceleration will be in the opposite direction of the motion. The cyclist is moving in a positive direction, so the cyclist’s acceleration must be in a negative direction.
Example 2: A Soccer Player
A soccer player is running backward (in the negative direction) to receive a pass from a teammate. Once they receive the ball, they quickly change direction and start running forward, speeding up as they go.
To determine the direction of acceleration for the soccer player, consider two parts of their motion:
- When the player is running backward in a negative direction: Since the player is slowing down and moving in a negative direction, the acceleration will be in the opposite direction of the motion (positive direction).
- When the player starts running forward in a positive direction: In this case, the player is speeding up while moving in a positive direction, so the acceleration will be in the same direction as the motion (positive direction).
In this scenario, the soccer player experiences positive acceleration in both stages, even though they are initially moving in a negative direction.
Review more examples with the Physics Classroom’s Acceleration Concept Builder.
Practice Using the Acceleration Formula in Word Problems
Now that we have reviewed acceleration and how to find both its magnitude and direction, we can apply this knowledge to solve more complicated physics word problems.
Example 1: Acceleration of a Falling Ball
A ball dropped from rest reaches a downward velocity of 24.5\text{ m/s} after falling for 2.5\text{ s}. What is the magnitude and direction of the ball’s acceleration?
To find the magnitude of the acceleration, we will substitute the given values into the formula for acceleration. Since the ball starts from rest, its initial velocity, v_i is 0\text{ m/s}.
a=\dfrac{\Delta v}{t}=\dfrac{v_f - v_i}{t}
a=\frac{24.5\text{ m/s}-0\text{ m/s}}{2.5\text{ s}}
a=9.8\text{ m/s}^2
Since the ball is speeding up, the acceleration will be in the same direction as the ball’s motion. The ball is moving downward in a negative direction. Therefore the direction of the acceleration is also downward or negative. In this case, this is the free fall acceleration and can be expressed as either -9.8\text{ m/s}^2 or 9.8\text{ m/s}^2 downward.
Example 2: Final Velocity of a Skateboarder
A skateboarder with an initial speed of 2\text{ m/s} accelerates down a ramp at a rate of 6\text{ m/s}^2 for 1.5\text{ s}. What is the skateboarder’s final velocity?
The first step to solve this problem is to use the formula for acceleration to determine the skateboarder’s change in velocity. Let’s substitute the values for acceleration and time into the formula:
a=\dfrac{\Delta v}{t}
6\text{ m/s}^2=\dfrac{\Delta v}{1.5\text{ s}}
Now, we can multiply both sides by the time to solve for the change in velocity:
\Delta v=(6\text{ m/s}^2)(1.5\text{ s})=9\text{ m/s}
Change in velocity is the difference between the object’s final and initial velocities. Using our change in velocity and the skateboarder’s initial velocity produces:
\Delta v = v_f - v_i
9\text{ m/s}=v_f-(2\text{ m/s})
The last step to solve for the final velocity is to add the initial velocity to the change in velocity:
v_f=9\text{ m/s}+2\text{ m/s}=11\text{ m/s}
Therefore, the skateboarder’s final velocity is 11\text{ m/s} directed down the ramp.
Kinematic Equations
Now that you know all four kinematic terms (time, displacement, velocity, and acceleration), you will be able to describe the motion of objects. For objects with uniform acceleration, the relationships between these variables are expressed through kinematic equations.
| Kinematic Equations v_f=v_i+at d=v_i t+\frac{1}{2}at^2 v_f^2=v_i^2+2ad d = \dfrac{v_i + v_f}{2}\cdot t |
The first kinematic equation is actually just a variation of the formula for acceleration solved for the final velocity. These equations will allow you to predict the motion of objects and solve for unknown variables in physics problems.
Conclusion
Acceleration is the rate of change of velocity and allows us to describe the motion of objects with changing velocities. It is important to remember that acceleration is a vector quantity with both magnitude and direction. To identify if an object’s acceleration is positive or negative, we need to consider both if the object is speeding up or slowing down and the direction the object is moving.
