Displacement Calculator with Steps | Physics & Engine

A displacement calculator finds how far, and in which direction, an object ends up from where it started: final position minus initial position.

A car moving at 5.0 m/s that speeds up at 2.0 m/s² for 4.0 s has a displacement of 36 m. For an engine, displacement is the volume all the pistons sweep: an 86.0 mm bore, 86.0 mm stroke four-cylinder displaces 1998 cc (2.0 L).

x and s in metres (m), u in m/s, a in m/s², t in seconds (s); engine bore b and stroke s in cm give V in cm³ (cc).

Rule of thumb: displacement can be smaller than the distance travelled, never larger. Walk 3 km north then 4 km east: the distance is 7 km but the displacement is only 5 km.

Displacement is the change in position of an object: where it ends minus where it started, with a direction. It is a vector, measured in metres (m). If you walk 10 m forward and 10 m back, you have moved 20 m, but your displacement is 0 m because you are back where you began.

1.1 How to calculate displacement in 4 steps

1. Pick a positive direction (forward, up, east) and write every value with a sign and a unit.
2. Convert to SI units: km/h ÷ 3.6 = m/s, cm ÷ 100 = m, minutes × 60 = s.
3. Choose the formula that matches what you know (table 1.2): end minus start, s = vt, or a SUVAT equation.
4. Substitute, solve and give a direction: s = (5.0)(4.0) + ½(2.0)(4.0)² = 36 m forward.

1.2 Displacement formula: which one to use

1.3 Distance vs displacement

Distance is the whole length of the path. Displacement is the straight line from start to finish, with a direction. They are equal only for a straight trip with no turning back.

1.4 Engine displacement formula (bore and stroke)

Engine displacement is the total volume swept by all the pistons: V = π/4 × bore² × stroke × cylinders. With bore and stroke in cm the answer is in cc; divide by 1000 for litres and by 16.387064 for cubic inches. An 86.0 mm × 86.0 mm four-cylinder is 1998 cc, sold as a 2.0 L engine.

1.5 Water displacement and density

A sinking solid pushes aside its own volume of water. Read the level before and after, subtract to get the volume, and divide the mass by it: 270 g ÷ 100.0 cm³ = 2.70 g/cm³, the density of aluminium.

1.6 Six calculators in one

A displacement calculator works this out for you, whatever information you have. This one has six ways in, because the word "displacement" is used in three different jobs:

• Motion equations (SUVAT). Enter any three of displacement s, initial velocity u, final velocity v, acceleration a and time t. The tool finds the other two, and the distance travelled when the object turns round.
• Between two points. Enter start and end coordinates in 1D, 2D or 3D. You get the components, the size of the displacement and its direction.
• Path: distance vs displacement. Enter each leg of a trip as a distance and compass bearing. You get the total distance and the net displacement side by side.
• Velocity function (calculus). Type v(t) such as 3t² − 12t + 9. The tool integrates it to give displacement and distance, and finds the turning points.
• Engine displacement. Enter bore, stroke and the number of cylinders to get cc, litres and cubic inches. Or solve backwards for the bore or the stroke.
• Water displacement density. Enter the mass and the water level before and after. You get the volume, the density, the closest material and the buoyant force.

A worked one-liner: a car at u = 5.0 m/s accelerates at a = 2.0 m/s² for t = 4.0 s. Then s = (5.0)(4.0) + ½(2.0)(4.0)² = 20 + 16 = 36 m, and v = 5.0 + 2.0 × 4.0 = 13 m/s.

Who uses it: students in CBSE and ICSE Class 9 to 12, JEE and NEET, IGCSE and GCSE, A level and AP Physics 1 meet displacement in the first chapter of mechanics. First-year university students use the calculus form. Mechanics, car buyers and engine builders use engine displacement in cc or cubic inches. Chemistry and physics labs use water displacement to measure the volume and density of odd-shaped solids, a method that goes back to Archimedes.

4.1 What displacement tells you

Displacement answers one question: how far from the start did the object end up, and in which direction? It ignores the route. A runner who laps a 400 m track once has run 400 m, but their displacement is 0 m, because they finish on the start line.

4.2 Is your answer a sensible size?

Compare it with everyday numbers. A person walks at about 1.4 m/s, so 100 m takes a little over a minute. A car at motorway speed (30 m/s) covers about 1 km in 33 s. A dropped stone falls 4.9 m in the first second and 44 m in 3 s. If your answer is 1000 times bigger or smaller than these, check for a km/h or a cm that was not converted.

4.3 The sign is the direction

In one dimension the sign of the displacement tells you the direction. Choose which way is positive before you start, and keep it. A ball thrown up at 20 m/s with up as positive has a = −9.8 m/s². After 3.00 s its displacement is +15.9 m: still above the hand, even though it is already falling.

4.4 Distance can be larger than displacement

Distance counts every metre moved. Displacement only counts start to finish. They are equal only when the object moves in a straight line without turning back. The thrown ball above rises 20.4 m and falls 4.5 m, so the distance is 24.9 m while the displacement is 15.9 m. This calculator gives both.

4.5 How the answer scales

From rest with constant acceleration, s = ½at². Double the time and the displacement becomes four times larger. Double the acceleration and it only doubles. For an engine, a 10% wider bore raises the displacement by 21%, but a 10% longer stroke raises it by just 10%.

4.6 What the v-t graph shows

On a velocity-time graph, the area between the line and the time axis is the displacement. Area above the axis is forward motion; area below counts as negative. The gradient of the line is the acceleration. This is why the calculus mode integrates v(t): integration is adding up area.

4.7 Engine displacement is a volume

For an engine, displacement means the swept volume: the space one piston moves through from top dead centre to bottom dead centre, times the number of cylinders. It is quoted in cc (cm³), litres or cubic inches. The badge on a car is usually rounded: a 1998 cc engine is sold as a 2.0 L.

4.8 Water displacement gives volume

When a solid sinks in water, the water level rises by exactly the solid's volume. Divide its mass by that volume and you have its density. Any density above 22.6 g/cm³ (osmium, the densest element) is a sign of a units mix-up.

▶ Press Calculate above to fill these five cards with your own answer.

This flowchart shows how a displacement calculator works, step by step, on the worked problem used across this page: a car moving at 5.0 m/s accelerates at 2.0 m/s² for 4.0 s. Find its displacement.

The same steps work for engine displacement: list bore, stroke and cylinders; convert to cm; check the bore is a diameter; apply V = π/4 × b² × s × n; and check by substituting back. For 86.0 mm × 86.0 mm × 4 that gives 1998 cc. Sections 9, 10, 11 and 14 do both by hand and in software.

1. Choose the type of problem. Pick a tab: Motion (SUVAT), Two Points, Distance vs Displacement, v(t) Calculus, Engine cc, Water Displacement or Batch.
2. Choose what to solve for. In the SUVAT and Engine tabs, tap the chip for the unknown. Its box turns into a dashed question mark.
3. Type your values. Fill the boxes you know. For SUVAT fill exactly three of s, u, v, a and t.
4. Pick a unit for each value. Every box has its own unit list, such as km/h, mph, ft, in or g. The calculator converts to SI units for you.
5. Set signs for direction. Choose which way is positive and give values in the opposite direction a minus sign.
6. Choose g and significant figures. Set g to match your exam board (9.80665, 9.81, 9.8 or 10 m/s²). Leave significant figures on Match the data unless your teacher says otherwise.
7. Or load a sample problem. Fourteen samples cover cars, constant velocity, free fall, a thrown ball, a walk, calculus, engines and a density trap. Nothing is calculated until you press Calculate.
8. Press Calculate. Or press Enter in any box. The answer appears with its unit, other units, two check badges and a model note.
9. Read the working. The GRESS table and the numbered steps show every conversion, equation and substitution, ready to copy into an exam answer.
10. Check the diagram and charts, then download. The live diagram and four charts are drawn from your numbers. Download Doc saves the full working as text; Download PDF prints it.

Excel has no DISPLACEMENT function, but each formula on this page is one line of arithmetic. Put one problem on each row, write the unit in every column header, and copy the formula down. The worked numbers are the same as Sections 7, 10 and 11: u = 5.0 m/s, a = 2.0 m/s², t = 4.0 s, and an 86.0 mm × 86.0 mm four-cylinder engine.

9.1 The functions you need

9.2 Step by step

9.3 The awkward cases

• Trig takes radians. =SIN(30) gives −0.988, not 0.5. Wrap degrees: =SIN(RADIANS(30)). Use =DEGREES(...) to turn an ATAN2 answer back into degrees.
• ATAN2 argument order. Excel is ATAN2(x, y). Python and R are atan2(y, x). Swapping them gives the complementary angle.
• Negative square. =-3^2 returns 9 in Excel, because the minus is applied first. Write =-(3^2) or keep signs inside brackets.
• Large and small numbers. Format the cell as Scientific (0.00E+00) to see 9.40E+11 m for an orbit. Excel keeps about 15 significant digits.
• Solving for time. When t is the unknown, use Data → What-If Analysis → Goal Seek: set E2 to your displacement by changing D2.

9.4 Why Excel might disagree with R, Python or this page

9.5 Charting it

1. Make a column of times (0, 0.5, 1.0 … 4.0 s) and a column of s with the SUVAT formula.
2. Insert → Scatter → Scatter with Smooth Lines. Never use the Line chart type: it spaces x values evenly whatever they are.
3. Add axis titles with units: "time t (s)" and "displacement s (m)".
4. For lab data, plot s against t² and add a linear trendline with its equation. The gradient is ½a.

9.6 Errors you will hit

This base-R script solves the same problems as the page: the car (36 m), two points (10 m at 53.13°) and the engine (1998.2 cc). It checks the answer with a second equation, works out the distance travelled, and saves one 300 dpi v-t graph with the displacement shaded.

10.1 The complete script

R, base only, saves displacement_vt.png

# ============================================================
# Displacement calculator in R: motion (SUVAT), two points,
# a walked path, and engine displacement, with one 300 dpi plot
# StatsUnlock - https://statsunlock.com/displacement-calculator/
# Base R only. No packages to install.
# ============================================================

# ---- 1. Your values (SI units: m, s, m/s, m/s^2) -----------
u <- 5.0     # initial velocity, m/s
a <- 2.0     # acceleration, m/s^2
t <- 4.0     # time, s

g <- 9.80665 # standard gravity, m/s^2 (use 9.81 or 9.8 to match your exam board)

# ---- 2. Motion: s = ut + 1/2 a t^2 and v = u + at ----------
s <- u * t + 0.5 * a * t^2
v <- u + a * t
cat(sprintf("Displacement s = %.4g m\n", s))
cat(sprintf("Final velocity v = %.4g m/s\n", v))

# check with a second equation: v^2 = u^2 + 2as
check <- abs(v^2 - (u^2 + 2 * a * s)) < 1e-9
cat("Check v^2 = u^2 + 2as:", ifelse(check, "OK", "FAILED"), "\n")

# distance travelled (differs from |s| if the object turns round)
t_turn <- if (a != 0) -u / a else NA
if (!is.na(t_turn) && t_turn > 0 && t_turn < t) {
  s_turn <- u * t_turn + 0.5 * a * t_turn^2
  distance <- abs(s_turn) + abs(s - s_turn)
} else {
  distance <- abs(s)
}
cat(sprintf("Distance travelled = %.4g m\n", distance))

# ---- 3. Displacement between two points --------------------
p1 <- c(2.0, 3.0)    # start (x, y) in m
p2 <- c(8.0, 11.0)   # end (x, y) in m
d <- p2 - p1
mag <- sqrt(sum(d^2))
theta <- atan2(d[2], d[1]) * 180 / pi   # degrees from +x
cat(sprintf("Two points: |d| = %.4g m at %.4g degrees\n", mag, theta))

# ---- 4. Engine displacement --------------------------------
bore_mm <- 86.0; stroke_mm <- 86.0; cylinders <- 4
cc <- pi / 4 * (bore_mm / 10)^2 * (stroke_mm / 10) * cylinders
cat(sprintf("Engine: %.1f cc = %.2f L = %.2f cubic inches\n", cc, cc / 1000, cc / 16.387064))

# ---- 5. One publication-ready figure: v-t graph ------------
tt <- seq(0, t, length.out = 100)
vv <- u + a * tt
png("displacement_vt.png", width = 6, height = 4, units = "in", res = 300)
par(mar = c(4.5, 4.5, 2.5, 1))
plot(tt, vv, type = "l", lwd = 3, col = "#2563eb", ylim = c(0, max(vv) * 1.1),
     xlab = "time t (s)", ylab = "velocity v (m/s)",
     main = sprintf("Area under the v-t line = displacement = %.4g m", s))
polygon(c(0, tt, t), c(0, vv, 0), col = adjustcolor("#2563eb", alpha.f = 0.22), border = NA)
invisible(dev.off())
cat("Saved displacement_vt.png\n")

Output when run: Displacement s = 36 m; Final velocity v = 13 m/s; Check OK; Distance travelled = 36 m; Two points: |d| = 10 m at 53.13 degrees; Engine: 1998.2 cc = 2.00 L = 121.94 cubic inches.

10.2 What each part does

R traps: trig functions use radians; round() goes half to even, so round(2.5) is 2; -2^2 is −4 because ^ binds tighter than the minus; log() is the natural log.

10.3 What the figure shows

The blue line is v = 5.0 + 2.0t. The shaded area under it is a trapezium of width 4.0 s and heights 5.0 and 13 m/s, so its area is ½(5.0 + 13)(4.0) = 36 m: the displacement. The title prints the same number the script calculates.

10.4 Common problems and fixes

• "object not found": run the script from the top, so u, a and t exist before they are used.
• No PNG appears: look in getwd(), the working directory, or give a full path to png().
• Wrong answer in km/h: convert first with u <- 72 / 3.6.

10.5 Useful one-liners

The Python version does everything the R script does and adds the calculus case: it integrates v(t) = 3t² − 12t + 9 with SciPy to get a displacement of 4 m and a distance of 12 m. It saves the same 300 dpi v-t graph.

Install once with pip install numpy scipy matplotlib.

11.1 The complete script

Python 3 with NumPy, SciPy and Matplotlib, saves displacement_vt.png

# ============================================================
# Displacement calculator in Python: motion (SUVAT), two points,
# v(t) integration and engine displacement, with one 300 dpi plot
# StatsUnlock - https://statsunlock.com/displacement-calculator/
# pip install numpy scipy matplotlib
# ============================================================
import math
import numpy as np
from scipy.constants import g          # 9.80665 m/s^2 (standard gravity)
from scipy.integrate import quad
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt

# ---- 1. Your values (SI units: m, s, m/s, m/s^2) -----------
u = 5.0   # initial velocity, m/s
a = 2.0   # acceleration, m/s^2
t = 4.0   # time, s

# ---- 2. Motion: s = ut + 1/2 a t^2 and v = u + at ----------
s = u * t + 0.5 * a * t**2
v = u + a * t
print(f"Displacement s = {s:.4g} m")
print(f"Final velocity v = {v:.4g} m/s")
print("Check v^2 = u^2 + 2as:", "OK" if math.isclose(v**2, u**2 + 2*a*s) else "FAILED")

# ---- 3. Displacement between two points --------------------
p1 = np.array([2.0, 3.0])    # start (x, y) in m
p2 = np.array([8.0, 11.0])   # end (x, y) in m
d = p2 - p1
mag = np.linalg.norm(d)
theta = math.degrees(math.atan2(d[1], d[0]))   # degrees from +x
print(f"Two points: |d| = {mag:.4g} m at {theta:.4g} degrees")

# ---- 4. Displacement and distance from v(t) ----------------
def vel(tt):
    return 3*tt**2 - 12*tt + 9          # m/s
disp, _ = quad(vel, 0, 4)
dist, _ = quad(lambda tt: abs(vel(tt)), 0, 4, points=[1, 3])
print(f"v(t) = 3t^2 - 12t + 9 from 0 to 4 s: displacement = {disp:.4g} m, distance = {dist:.4g} m")

# ---- 5. Engine displacement --------------------------------
bore_mm, stroke_mm, cylinders = 86.0, 86.0, 4
cc = math.pi / 4 * (bore_mm / 10)**2 * (stroke_mm / 10) * cylinders
print(f"Engine: {cc:.1f} cc = {cc/1000:.2f} L = {cc/16.387064:.2f} cubic inches")
print(f"(g used elsewhere: {g} m/s^2)")

# ---- 6. One publication-ready figure: v-t graph ------------
tt = np.linspace(0, t, 100)
vv = u + a * tt
fig, ax = plt.subplots(figsize=(6, 4))
ax.plot(tt, vv, color="#2563eb", lw=3)
ax.fill_between(tt, 0, vv, color="#2563eb", alpha=0.22)
ax.set_xlabel("time t (s)")
ax.set_ylabel("velocity v (m/s)")
ax.set_title(f"Area under the v-t line = displacement = {s:.4g} m")
ax.set_ylim(0, vv.max() * 1.1)
fig.savefig("displacement_vt.png", dpi=300, bbox_inches="tight")
print("Saved displacement_vt.png")

Output when run: Displacement s = 36 m; v = 13 m/s; Check OK; Two points: |d| = 10 m at 53.13 degrees; v(t) displacement = 4 m, distance = 12 m; Engine: 1998.2 cc = 2.00 L = 121.94 cubic inches.

11.2 What each part does

Python traps: math.sin takes radians; round(2.5) is 2 (half to even); floats hold about 15 to 16 digits; format significant figures with f"{x:.3g}".

11.3 What the figure shows

The same trapezium as the R figure: area 36 m under a line rising from 5.0 to 13 m/s. Change u, a or t and the shaded area and the title change together.

11.4 Common problems and fixes

• ModuleNotFoundError: run pip install numpy scipy matplotlib in the same environment as Python.
• Distance equals displacement when it should not: pass the turning points to quad with points=.
• No window opens: the script saves a file instead of showing a window. Look for displacement_vt.png.

11.5 Useful one-liners

Run it in a terminal (python displacement.py), in Jupyter, or paste it into Google Colab.

12.1 Constants and exact conversions

Source: NIST CODATA 2022 and the SI Brochure for g₀; IAU Resolution B2 (2012) for the astronomical unit.

12.2 Length and speed conversions

12.3 Typical displacements and speeds

12.4 Engine displacement units and typical sizes

Sizes are swept volume from the formula V = π/4 × b² × s × n. Makers often round the badge figure.

12.5 Densities for the water displacement method

Room-temperature values, rounded. Glass, granite and wood vary with composition.

12.6 SI prefixes you will meet

This is the written method examiners expect, worked on the same example as Sections 7, 9, 10 and 11: a car moving at 5.0 m/s accelerates at 2.0 m/s² for 4.0 s. Find its displacement and final velocity.

14.1 The written method (GRESS)

1. Given. u = 5.0 m/s, a = 2.0 m/s², t = 4.0 s. Forward is positive.
2. Required. s (displacement, m) and v (final velocity, m/s).
3. Equation. s = ut + ½at² (it has no v), then v = u + at.
4. Substitute, with units. s = (5.0 m/s)(4.0 s) + ½(2.0 m/s²)(4.0 s)². v = 5.0 m/s + (2.0 m/s²)(4.0 s).
5. Solve. s = 20 + 16 = 36 m. v = 5.0 + 8.0 = 13 m/s.
6. Check the units and the size. (m/s)(s) = m and (m/s²)(s²) = m. Check: v² = 169 and u² + 2as = 25 + 144 = 169. Both agree.
7. Write the answer. s = 36 m forward, v = 13 m/s (2 significant figures, matching the data).

Engine version by hand: bore 86.0 mm = 8.60 cm, stroke 8.60 cm. Area = π/4 × 8.60² = 58.09 cm². One cylinder = 58.09 × 8.60 = 499.6 cm³. Four cylinders = 1998 cc = 2.0 L.

14.2 On a scientific calculator

Check the screen shows D (degrees) before any angle work. tan⁻¹(Δy/Δx) only gives the right angle when Δx is positive; add 180° when Δx is negative. Type each line in one go instead of rounding in the middle.

14.3 Measure it yourself: two school lab practicals

The gradient of s against t² is about 0.50 m/s², so a ≈ 1.0 m/s². In the Batch tab enter each row as 0, 1.0, t to compare predicted and measured s.

14.4 Practice problems with answers

14.5 Mistakes that lose marks

• Giving a distance when the question asks for displacement, or the other way round.
• Leaving speeds in km/h. Divide by 3.6 before using any equation.
• Dropping the minus sign on a deceleration or on a downward displacement.
• Adding the sizes of two perpendicular displacements (3 + 4 = 7) instead of using Pythagoras (5).
• Using the radius instead of the diameter in the engine formula, which gives an answer four times too big.
• Forgetting the direction of a 2D displacement. A vector needs a size and a direction.
• Writing a final answer with no unit, or with more significant figures than the data.

Use the SUVAT equations when:

• ✓ The acceleration is constant (including free fall with no air resistance).
• ✓ The motion is along one straight line.
• ✓ You know three of s, u, v, a and t.

Use something else when:

• ✗ The acceleration changes: integrate v(t), or find the area under a measured v-t graph.
• ✗ The motion is in 2D or 3D with known positions: subtract the coordinates (Two Points mode).
• ✗ The trip has several straight legs: add them as vectors (Distance vs Displacement mode).
• ✗ Air resistance matters, as for a skydiver: SUVAT overestimates the fall. Use a drag model.
• ✗ You need the volume an engine sweeps: that is engine displacement, a volume, not a length.

Real examples:

1. Braking distance of a car from a speed and a deceleration: SUVAT, v² = u² + 2as.
2. A hiker's GPS track of several legs: path mode, distance vs displacement.
3. A robot arm moving between two points in 3D: two-points mode.
4. A data logger that gives v(t) as a formula: calculus mode.
5. Buying a motorbike and comparing 125 cc with 150 cc: engine mode.
6. Identifying a metal in a school lab: water displacement density mode.

Assumptions

1. Constant acceleration (SUVAT mode). Check that a does not change during the motion. If a really falls with time, as for a car easing off, SUVAT with the starting a overestimates the displacement.
2. No air resistance. Check that speeds are low or objects are dense. When drag matters, SUVAT overestimates the displacement of a falling object: 490 m against about 340 m for a skydiver after 10 s.
3. Straight-line motion in SUVAT. Check that the path does not curve. For curved paths, use the vector modes.
4. Flat ground for paths. Legs are added on a flat map. Over hundreds of kilometres the curved Earth makes the flat-map displacement slightly too large.
5. Perfect cylinders for engines. The formula gives swept volume only. It ignores the combustion chamber, so it underestimates the total cylinder volume, and badge figures are rounded.
6. Fully submerged, non-absorbing solid for density. Air bubbles make the volume too large and the density too small. A solid that soaks up water makes the volume too small and the density too large.
7. Exact numerical integration. Simpson's rule is exact for polynomials up to cubic. For sharp or very fast-changing functions it may carry a tiny error in the last digit.

Limitations

• It does not model drag. For falling objects with air resistance, a drag model is needed.
• It does not do relativistic motion. At speeds near light speed (above about 0.1c) Newtonian kinematics fails.
• It does not handle rotating reference frames or the Coriolis effect.
• The engine mode assumes 100% volumetric efficiency for air flow. Real engines fill about 80 to 95% of their swept volume.
• The density match only suggests a material. Alloys and mixtures can share a density.
• It is not for single displacement reactions in chemistry. Those are about atoms swapping places, not about distance; see the stoichiometry tool in Section 21.

What the answer is

A displacement calculator gives the straight-line change in position, with its direction, in metres. For the worked car problem it is 36 m forward. For an engine it gives a volume instead: 1998 cc for the 86.0 mm × 86.0 mm four-cylinder. For a solid in water it gives the volume that leads to density: 2.70 g/cm³ for the aluminium block.

What it means physically

Displacement compares only the start and the end. That is why it can be zero after a long journey, and why it is always shorter than or equal to the distance. The difference between the two tells you how much the path bent or doubled back. On a velocity-time graph the displacement is the net area under the line, which is also what integration measures.

Watch out for

Units (km/h, mm, grams typed as kilograms), signs (pick a positive direction), and the g your exam board uses. For engines, remember the bore is a diameter. For angles, use atan2 or check the quadrant.

Next steps

Divide displacement by time to get velocity with the velocity calculator, then find how quickly that velocity changes with the acceleration calculator. Fit a lab s-t² graph with linear regression. Come back to this displacement calculator whenever you need a displacement, a distance or an engine size with every step shown.

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