Displacement Calculator

Displacement Calculator returns the change in position from any three of initial velocity, final velocity, acceleration and time. It also adds up velocity legs and N, S, E, W legs.

What the Displacement Calculator Solves

Displacement is the change in position of an object, written Δx = x − x₀. It has a size and a direction, so it can be positive, negative or zero. The Displacement Calculator finds it from the values a problem actually gives, with no formula to rearrange by hand.

Four of the Known Values settings cover motion with constant acceleration. Each one takes three of the four kinematic quantities, which are initial velocity, final velocity, acceleration and time. The fourth quantity is worked out and shown on the first result card.

Start and End Position is the direct route when a problem gives positions. It subtracts the start from the end, and a turning point lets it report the distance traveled as well. Velocity × Time Legs adds up to four stretches of steady velocity along one line.

N, S, E, W Legs handles motion in two dimensions. It adds up to five distances, each with a compass direction or a custom angle, and returns the straight line from start to finish with its direction.

Velocity can be entered in m/s, km/h, ft/s or mph, and acceleration in m/s², ft/s² or g. One g is the standard gravity of 9.80665 m/s². Lengths convert with 1 ft = 0.3048 m and 1 mi = 1,609.344 m, the exact factors in the NIST conversion tables.

The tool sits with the other motion tools under Kinematics Calculators. A launch speed in ft/s, such as the output of the Arrow Speed Calculator, can go straight into the initial velocity field without converting it first.

Displacement Formula for Each Known Set

All four constant-acceleration formulas come from the two kinematic equations in OpenStax University Physics, v = v₀ + at and x = x₀ + v₀t + ½at². The displacement calculator formula changes with the setting, and the result panel names the one it used.

Start and end position

This is the definition itself, and it needs no velocity or acceleration. A turning point between the two readings changes the distance traveled but never the displacement. Entering the time taken adds the average velocity.

$$\Delta x = x – x_0$$

Initial velocity, acceleration and time

This is the standard form for an object that speeds up or slows down for a known time. Braking and any other backward acceleration go in with a minus sign. The final velocity is returned as well, from v = v₀ + at.

$$\Delta x = v_0 t + \frac{1}{2} a t^2$$

Initial velocity, final velocity and time

With constant acceleration the average velocity sits midway between the start and end values. Multiplying that average by the time gives the displacement. This is how to find displacement with v and t when a problem gives no acceleration.

$$\Delta x = \frac{1}{2} (v_0 + v) t$$

Both velocities and acceleration, no time

Some problems give both velocities and the acceleration but never mention time, and braking distance is the usual case. The calculator returns the time taken too, found from t = (v − v₀) ÷ a.

$$\Delta x = \frac{v^2 – v_0^2}{2a}$$

Final velocity, acceleration and time

When only the final velocity is known, the formula runs backward from the end of the motion. The half-acceleration term changes sign as a result. The initial velocity is returned from v₀ = v − at.

$$\Delta x = v t – \frac{1}{2} a t^2$$

Velocity legs and compass legs

For steady stretches the displacement is the sum of velocity × time over every leg, with backward legs counted as negative. For compass legs, each distance is split into a north part and an east part, and the two totals N and E give the resultant R.

$$\Delta x = v_1 t_1 + v_2 t_2 + v_3 t_3 + v_4 t_4$$

$$R = \sqrt{N^2 + E^2}$$

Acceleration Displacement Examples

The table runs four cases through the tool as an acceleration displacement calculator. The first two are worked examples from the OpenStax chapter linked above. The textbook rounds the dragster to 402 m and gives 10.0 m/s for the airplane’s final velocity.

The dragster covers 0.2497 mi, almost exactly a quarter mile, and the unit card shows that conversion beside the result in feet and kilometers. The car’s 64.29 m is its stopping distance, because the final velocity is zero.

Displacement grows with the square of the time whenever acceleration is not zero. Starting at 12 m/s and gaining 3 m/s² for 5 s gives 97.5 m, yet only 39.38 m of it, or 40.4%, comes in the first half of the time. The halfway point of the displacement is reached at 2.96 s.

These formulas hold only while acceleration stays constant. A bullet in flight loses speed to drag at a changing rate, which is why its slowdown is described with a ballistic coefficient, as in the Ballistic Coefficient Calculator.

Distance vs Displacement After a Turnaround

Distance counts every meter of the path, and displacement counts only the net change in position. The two match in one-way motion and split apart the moment an object reverses. The tool then works as a distance displacement calculator and reports both.

Take an initial velocity of 12 m/s with an acceleration of −3 m/s² for 10 s. The object moves 24 m forward, stops at 4 s and runs back past its start point. Its displacement is −30 m, yet the distance traveled is 78 m.

The result cards change to suit the reversal. They show the turnaround time, the farthest point and the 8 s it takes to be back at the start. Average speed over the path is 7.8 m/s, while average velocity is −3 m/s.

Total displacement for a trip in stages comes from Velocity × Time Legs. Moving at 5 m/s for 60 s and then back at −2 m/s for 30 s gives 300 − 60 = 240 m of displacement over 360 m of path. A final velocity from the Conservation of Momentum Calculator can be entered as a leg in the same way.

That trip has an average velocity of 2.67 m/s and an average speed of 4 m/s. The path efficiency card reads 66.7%, the share of the path that counted as net progress. A round trip that ends where it began scores zero displacement however far it went.

Displacement Calculator for North South East West

Displacement in two dimensions is found by components. Each leg is split into a north part and an east part, with south and west counted as negative. The parts are summed, and the Pythagorean theorem gives the length of the straight line from start to finish.

An OpenStax College Physics example walks 9 blocks east and 5 blocks north. Entered as 9 km east and 5 km north, the tool returns 10.3 km at 29.1° north of east, matching the textbook’s 10.3 blocks and 29.1°. The walk was 14 km, so the straight line is 3.7 km shorter.

Direction is given two ways. The angle from the east–west line is the form physics textbooks use, and the compass bearing runs clockwise from north. The same example has a bearing of 060.9° and a return bearing of 240.9°.

Textbook problems often give a leg as an angle, such as 30° north of east. Choosing Custom angle in a leg’s direction menu opens an angle field with eight quadrant references and a 0 to 360° bearing. A 10 km leg at 30° north of east splits into 5 km north and 8.66 km east.

Adding a second leg of 6 km at 20° west of north brings the totals to 10.64 km north and 6.61 km east. The displacement is 12.52 km at 58.2° north of east, against 16 km of path, and the step list shows the north and east part of each leg.

Diagonal legs work too. The default trip of 4 km north, 3 km northeast and 2 km east sums to 6.12 km north and 4.12 km east. Its displacement is 7.38 km at 56° north of east, against 9 km walked, and the path diagram marks it as a dashed line.

The car’s horizontal travel in the Car Jump Distance Calculator is a displacement of the same kind. It is measured in a straight line from the ramp lip to the landing point, not along the curved flight path.

Engine, Ship and Current Displacement

The word displacement has other meanings that share nothing with a change in position. A search for an engine displacement calculator is asking about cylinder volume, and ship displacement is a weight. The tool above is a displacement calculator for physics motion problems only.

How do I calculate engine displacement in cc?

Engine displacement is the volume swept by all the pistons in one stroke. The cc formula multiplies the bore area by the stroke S and by the number of cylinders N, with the bore B and the stroke both in centimeters.

$$V = \frac{\pi}{4} \times B^2 \times S \times N$$

A four-cylinder engine with a 7.4 cm bore and a 7.55 cm stroke gives 0.7854 × 54.76 × 7.55 × 4 = 1,299 cc. A figure in cubic inches converts to cc when multiplied by 16.387, because one inch is 2.54 cm.

What does 1300cc mean?

It means the cylinders sweep a combined 1,300 cubic centimeters, which is 1.3 liters or about 79.3 cubic inches. The engine in the example above, at 1,299 cc, would be sold as a 1300cc or 1.3 L engine.

What does a ship displacement calculator work out?

Ship displacement is the weight of the water a floating hull pushes aside. By Archimedes’ principle that weight equals the weight of the ship itself, so it is quoted in tonnes and has nothing to do with meters moved.

Is displacement current the same formula?

No, displacement current belongs to electromagnetism. It is the term Maxwell added to Ampere’s law for a changing electric field. It equals the permittivity of free space times the rate of change of electric flux, and no object changes position.

$$I_d = \varepsilon_0 \frac{d\Phi_E}{dt}$$

Displacement in Physics Explained

How do I calculate displacement?

Subtract the starting position from the final position. An object that moves from x = 2 m to x = 9 m has a displacement of +7 m, and one that moves from 9 m back to 2 m has a displacement of −7 m.

The Start and End Position setting does this and adds the path. A trip from 2 m out to 12 m and back to 5 m in 20 s has a displacement of 3 m over a 17 m path. Average velocity is 0.15 m/s, average speed is 0.85 m/s, and path efficiency is 17.6%.

When positions are not given, use the motion instead. Pick the formula that matches the known values, which is what the Known Values menu in the Displacement Calculator does. A start position can be entered so the result also shows the final position.

How do I convert velocity to displacement?

Multiply the velocity by the time. A steady 25 m/s held for 12 s gives 300 m. If the velocity changes at a constant rate, use the average of the start and end values, so 10 m/s rising to 30 m/s over 12 s gives 20 × 12 = 240 m.

On a velocity–time graph the displacement is the area under the line. Area below the time axis counts as negative, which is how a return leg cancels part of an outbound one in the total.

How do you calculate average displacement?

Displacement is a single net change, so it is not usually averaged. The question normally means average velocity, which is displacement divided by the time taken. A displacement of 240 m in 90 s is an average velocity of 2.67 m/s.

What are the different types of displacement?

Mechanics uses two. Linear displacement is a change in position along a line or across a plane, measured in meters. Angular displacement is the angle an object turns through, measured in radians.

Linear displacement is then described by its dimensions. A one-dimensional problem uses a plus or minus sign for direction, and a two-dimensional problem needs a magnitude with an angle or bearing. The Displacement Calculator covers both linear cases.

What are examples of displacement?

A sprinter who runs a straight 100 m has a displacement of 100 m. A runner who completes one lap of a 400 m track has covered 400 m of distance with zero displacement, because the finish is the start.

An elevator that rises 12 m and then drops 4 m has a displacement of +8 m after 16 m of travel. A hiker who goes 3 km east and then 4 km north ends 5 km from camp, at 53.1° north of east.