Displacement Calculator

Displacement is the amount and direction of change in an object's position from its starting point. It is a straight line distance between the start and end points and can be expressed in various units such as meters, feet or miles.

This calculator will determine the displacement in four different ways depending on what information is available. You can input a constant velocity and time, initial and final velocities or acceleration. Alternatively you can enter a change in velocity without specifying a time. In these cases the calculator will also calculate the average velocity, final velocity or corresponding time in addition to the displacement.

Displacement is not the same as distance.

Distance is the entire length of the path traveled by an object. Displacement on the other hand, is the straight line distance between the starting and ending point and also includes direction.

For example, if you drive five kilometres to work and then another five kilometres back home, the total distance traveled is ten kilometres. However, since your final location is the same as where you started from, the displacement is zero.

As the displacement has a direction, it is a vector and can have negative values. A negative value simply means that the object ends up on the opposite side of its starting point. The displacement can never be larger than the distance traveled; at most they are equal. This only applies if the motion always happens in a straight line though.

Four equations for displacement.

These four equations all derive from the equations of uniformly accelerated motion, which are commonly referred to as SUVAT equations. As each equation uses a different combination of known quantities, a calculation tool will first ask what information is known.

1. Constant speed:

If the speed is constant then displacement is equal to velocity times time.

A car traveling at a speed of 20 meters per second for 5 seconds will cover a distance of 100 meters, which is the product of 20 and 5. This is an equation used in everyday road traffic. If a vehicle travels at a speed of 70 miles per hour for 2 hours, then the displacement is 140 miles.

2. From initial and final speed.

If the speed is changing constantly, then the average speed will be exactly halfway between the initial and final speeds, and the distance traveled will be that average speed multiplied by the time.

In this case u is the initial velocity and v is the final velocity. If a sprinter accelerates from an initial velocity of 20 meters per second to a final velocity of 23 meters per second in 45 seconds, then the distance traveled is half of the product of 43 and 45, which equals 967.5 meters. The formula can also be rearranged to find time given a known distance: t = 2d / (u + v).

3. From acceleration:

If the initial velocity, acceleration and time are known, then the distance can be calculated by adding the constant speed portion to the accelerated portion.

If an object is released from rest it will fall due to gravity at a rate of about 9.81 meters per second squared. The distance fallen after two seconds would be half the product of 9.81 and 4, or about 19.6 meters. You can use Earth's acceleration directly by entering "g" in the acceleration field.

4. When time is unknown.

Sometimes even when speed and acceleration are known it is not clear how long the change in speed lasts for. Here is a formula for braking distance that does not require time.

Suppose a car is moving at a speed of 30 meters per second and comes to rest with retardation of 6 meters per second squared. The distance covered by the car before coming to rest is 75 meters, which is equal to 900 divided by 12. The calculator also gives out the time taken for stopping, which is (v minus u) divided by a.

To use this calculator:

Start with the main menu and select the scenario that matches your question. The input fields will change accordingly so that only the measurements required for this scenario are shown.

If you enter known values, the calculator will instantly show you the distance traveled. Each field has a unit conversion option so that for example you can enter speed in miles per hour and time in hours to get the distance traveled in miles. All conversions are done by the calculator.

If the motion is going backwards or an object is slowing down, leave the sign of the velocity or acceleration alone. This sign will keep track of direction until you get to your final answer.

Examples:

These examples use integers so that you can verify the calculations manually.

A car traveling at a constant speed and a car traveling with average speed both travel the same distance of 50 meters. The average speed during acceleration from 0 to 20 meters per second is 10 meters per second, which is the same as a constant speed of 10 meters per second.

Average and final speed:

If the calculation tools include time data, average speed is also given. This corresponds to a single constant speed that would cover the same distance in the same amount of time. Average speed is distance divided by time.

In acceleration scenarios, the final velocity is also stated. This is the speed of the object at the end of its motion and is calculated based on the formula v = u + a t. These additional results allow for a better understanding of a problem involving motion.

Units of Displacement

Since the displacement is a type of length, any unit of length can be used. The meter is the SI unit, while miles are convenient for road trips and feet are good for working on cars.

Why displacement is important

Displacement is the starting point for nearly all motion problems in physics. It's used to calculate where a projectile will land, how far a car travels before coming to rest, and how high an elevator goes between two floors.

Engineers use displacement to calculate braking distances and safe following distances, navigators plot straight-line routes between two points, and animators move objects smoothly across the screen. In all of these cases it is important to consider the net change in position, not the winding path taken to get there.

Common Mistakes

The most common mistake is to confuse displacement with distance. If an object reverses direction during its motion, these two values are different; only the linear change in position represents the displacement.

Be careful about the sign of velocity and acceleration. A car that is slowing down has a negative acceleration relative to its direction of motion. If you get the sign wrong, it will change the sign of your answer. Make sure you use consistent units or do the conversion yourself. Don't mix miles per hour with seconds.

This tool is for general educational purposes and solving everyday problems. For technical applications, safety issues or other risky activities verify the values against the standards and data required by your project.