Physics Displacement Formula: How to Calculate Displacement

By: Sascha Bos  | 
In physics, displacement refers to a change in an object's position. If you drive 1 mile (1.6 kilometers) down the road to your friend's house, your car's displacement is 1 mile (1.6 kilometers). Marin Tomas / Getty Images

Physicists use the displacement formula to find an object's change in position. It sounds simple, but calculating displacement can quickly get complicated.

Let's break it down using some solved examples.


What Is Displacement?

In physics, displacement refers to an object's change in position. For example, if you walk 1 mile (1.6 kilometers) down the street to your friend's house, your displacement is 1 mile (1.6 kilometers).

But if you hike a 1-mile (1.6-kilometer) loop that starts and ends at the same point, your displacement is actually 0. And if your friend's house is 1 mile (1.6 kilometers) away, but you take a meandering route rather than walking in a straight line, your total displacement is still 1 mile (1.6 kilometers).


That's because displacement measures the shortest distance between an object's initial position and its final position. When it comes to displacement, the actual path the object takes does not matter; displacement always refers to the shortest path possible.

Displacement vs. Distance: What's the Difference?

Distance refers to the total distance covered by an object, whereas displacement is the object's change in position. Wait, aren't those the same thing?

Only if the object travels the shortest path from point A to point B.

So, if you hike a 1-mile (1.6-kilometer) loop, your distance is 1 mile (1.6 kilometers), but your displacement is 0. Note that displacement is never greater than distance, because displacement represents the shortest distance possible.

In science-speak, distance is a scalar quantity, like speed, while displacement is a vector quantity, like velocity. Scalar quantities only have magnitude, while vector quantities have both magnitude and direction.


Displacement Formulas

The basic formula to calculate displacement is a reworking of the velocity formula:

d = vt

Where d is displacement, v is average velocity, and t is the time period, or the time it took to get from point A to B.


If the object has constant velocity, solving for displacement is straightforward. If not, you can use the initial and final velocities to calculate displacement using the acceleration formula. (And if you have multiple velocities, you may want to use a displacement calculator like the ones provided by Omni Calculator and Calculator Soup.)

Using the Acceleration Formula

If the object you want to calculate displacement for has constant acceleration, you can use the acceleration formula, based on Newton's third law of motion:

a = (v1 – v)/t

Where a is acceleration, v1 is the object's initial velocity, v is the object's final velocity and t is time. Using this formula to calculate displacement, you would have:

d = 1/2(at2)+vt

Where d is displacement, a is acceleration, t is the time it took to get from the start point to the end point, and v is the final velocity.


Displacement Formula Example

Let's calculate displacement using some real-world examples.

Example 1: Constant Velocity

If an object has constant velocity, you can use the velocity formula to calculate displacement. For example, f a car has an average velocity of 25 miles per hour (40 kilometers per hour) and travels for 15 minutes, what is its displacement?


To solve this, we can use the formula:

d = vt

Where v is 25 miles per hour (40 kilometers per hour) and t is 15 minutes, or 0.25 hours.

d = (25)(0.25) or d = (40)(0.25)
d = 6.25 miles (10 kilometers)

So, after traveling 25 minutes at a constant velocity of 25 miles per hour (40 kilometers per hour), this car has a displacement of 6.25 miles (10 kilometers).

Example 2: Constant Acceleration

If a car traveling at 22.4 miles per hour (36 kilometers per hour) needs to come to a stop within 30 seconds, what would the vehicle's displacement be?

First, we can calculate the acceleration.

a = (v1 – v)/t

Where v1 is the car's initial velocity (22.4 miles per hour or 36 kilometers per hour), v is the car's final velocity (0 miles per hour), and t is the time (30 seconds).

First, let's do some unit conversion to calculate the acceleration in SI units — meters per seconds squared.

v1 = 36 kilometers per hour
v1 = 36,000 meters per hour
v1 = 10 meters per second

Then we can plug that into our formula:

a = (v1 – v)/t
a = (10 – 0)/30
a = 10/30
a = 0.33 meters per second squared

Now that we know the acceleration, we can calculate the displacement.

d = 1/2(at2)+vt

Where a is acceleration (0.33 meters per second squared), t is time (30 seconds), and v is final velocity (0 meters per second).

d = 1/2(0.33 x 30)2 + (10 x 30)
d = 1/2(9.9)2 + (300)
d = 1/2(98.01) + (300)
d = 49.005 + 300
d = 349.005 meters

The car had a displacement of 349.005 meters (1,145 feet) as it came to a stop.