Distance Calculator

What Is Distance?

Distance is the total length of the path an object travels, regardless of direction. If you walk 3 metres forward and 3 metres back, the total distance is 6 metres, even though you end up where you started.

Distance is a scalar quantity: it has magnitude only and is always ≥ 0. The SI unit is the metre (m).

Distance Formula

d = s \times t

For constant speed, distance equals speed multiplied by time. This is the most commonly used distance formula for everyday calculations like travel planning.

Distance\u2013Time Graphs

On a distance–time graph, constant speed produces a straight line. Acceleration produces a curve (parabola for constant acceleration). The slope at any point equals the instantaneous speed.

Distance vs Displacement

Property	Distance	Displacement
Type	Scalar	Vector
Can be zero after motion?	No	Yes (round trip)
Can be negative?	No	Yes
Path-dependent?	Yes	No (start to finish)
Relation	Distance ≥ \|displacement\|. Equal only for straight-line, one-direction motion.

Distance with Constant Acceleration

With initial velocity

\Delta x = v_0 t + \tfrac{1}{2}at^2

No time needed

\Delta x = \frac{v_f^2 - v_0^2}{2a}

When acceleration is constant, distance grows quadratically with time. The first formula is useful when you know time; the second when you know initial and final velocities.

Coordinate Distance (Euclidean)

2D Distance

d = \sqrt{(\Delta x)^2 + (\Delta y)^2}

3D Distance

d = \sqrt{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2}

The Euclidean distance formula gives the straight-line distance between two points in 2D or 3D space. It is derived from the Pythagorean theorem.

Circular Distance (Arc Length)

s = r\theta

The arc length of a circular path equals the radius times the central angle (in radians). One full revolution (θ = 2π) gives the circumference C = 2πr.

Applications: wheel travel per revolution, track curves, circular orbits, pendulum arc, gear teeth spacing.

Great-Circle Distance (Haversine)

d = 2R \arcsin\!\left(\sqrt{\sin^2\!\frac{\Delta\phi}{2} + \cos\phi_1\cos\phi_2\sin^2\!\frac{\Delta\lambda}{2}}\right)

The Haversine formula calculates the shortest distance between two points on a sphere given their latitudes (φ) and longitudes (λ). It uses Earth’s mean radius R = 6,371 km.

This “as the crow flies” distance is always shorter than the actual travel distance by road. Accuracy is within ~0.5% for most locations; for centimetre precision, use WGS-84 ellipsoidal formulas (Vincenty).

Braking Distance

d_{brake} = \frac{v_0^2}{2|a|}

Braking distance depends on the square of speed. Doubling your speed quadruples the stopping distance. This is critical for road safety.

Speed	Braking (dry, a = 8 m/s²)	Braking (wet, a = 5 m/s²)
30 km/h (8.3 m/s)	4.3 m	6.9 m
50 km/h (13.9 m/s)	12.1 m	19.3 m
80 km/h (22.2 m/s)	30.9 m	49.4 m
100 km/h (27.8 m/s)	48.2 m	77.2 m
120 km/h (33.3 m/s)	69.4 m	111.1 m

Values are braking distance only (not including reaction time). Total stopping distance = reaction distance + braking distance.

Real-World Distances

Reference	Distance
Marathon	42.195 km
London to Paris	344 km (direct)
New York to Los Angeles	3,944 km (direct)
Earth circumference	40,075 km
Earth to Moon	384,400 km
Earth to Sun	149.6 million km (1 AU)
1 light-year	9.461 × 10¹² km

How to Use the Calculator

Select a mode: constant speed, acceleration, kinematic, coordinate, or multi-leg.
Enter values with units.
Click Calculate.
Review the distance, formula breakdown, and unit conversions.

Example Calculations

Car: 100 km/h for 2.5 hours

d = 27.78 × 9,000 = 250,000 m = 250 km

Free fall: 4 seconds from rest

d = ½ × 9.81 × 16 = 78.5 m

Braking: 100 km/h, a = −8 m/s²

d = 27.78² / (2 × 8) = 48.2 m

2D: (1, 2) to (4, 6)

d = √(9 + 16) = √25 = 5 m

Common Mistakes

Confusing distance (scalar) with displacement (vector).
Using d = st when acceleration is present (should use d = v₀t + ½at²).
Forgetting that braking distance scales with speed squared, not linearly.
Mixing units (km with seconds, or miles with metres) without converting.
Not accounting for reaction distance in total stopping distance.

Accuracy and Limitations

Distance formulas assume constant speed or constant acceleration. Real motion involves variable speed and acceleration. Braking distances are theoretical values for flat, straight roads and do not include driver reaction time (typically 0.7–1.5 s). The coordinate distance is Euclidean (straight-line); actual travel distances along roads are longer. This tool is for education and estimation.

FAQ

What is distance in physics?›

Distance is the total length of the path travelled by an object. It is a scalar quantity (no direction) and is always ≥ 0. The SI unit is the metre (m).

What is the difference between distance and displacement?›

Distance is the total path length (always positive). Displacement is the straight-line change in position from start to finish (can be zero or negative). For a round trip, distance > 0 but displacement = 0.

How do you calculate distance from speed and time?›

Multiply speed by time: d = s × t. Make sure both are in compatible units (e.g., m/s and seconds, or km/h and hours).

How does acceleration affect distance?›

With constant acceleration starting from rest, d = ½at². With initial velocity, d = v₀t + ½at². Distance grows quadratically with time when accelerating.

What is braking distance?›

Braking distance is how far a vehicle travels while decelerating to a stop. From v² = v₀² + 2ad with v = 0: d = v₀² / (2|a|). It scales with the square of speed.

Why does doubling speed quadruple braking distance?›

Braking distance d = v²/(2a). If v doubles, v² quadruples, so d quadruples. At 60 km/h the braking distance is 4 times that at 30 km/h.

How do you find distance between two points?›

In 2D: d = √((x₂−x₁)² + (y₂−y₁)²). In 3D: add (z₂−z₁)² inside the square root. This is the Euclidean distance formula.

What is total distance for a multi-leg journey?›

Sum all individual segment distances: d = d₁ + d₂ + d₃ + … Distance is additive regardless of direction.

Can distance be negative?›

No. Distance is the magnitude of path length and is always ≥ 0. Displacement can be negative (indicating direction), but distance cannot.

How far does a car travel at 100 km/h for 1 hour?›

d = 100 km/h × 1 h = 100 km. In metres: 100,000 m. In miles: about 62.1 miles.

What is arc length?›

Arc length is the distance along a circular path. It equals radius times central angle in radians: s = rθ. For a full circle, s = 2πr (the circumference).

How do you calculate the distance between two GPS coordinates?›

Use the Haversine formula, which gives the great-circle (shortest) distance on a sphere. It uses latitudes, longitudes, and Earth’s radius (6,371 km). The result is the “as the crow flies” distance.

Sources

Author & technical reviewer

Manish Kumar

PhysicsCalcs tools are reviewed with an educational focus: clear formulas, transparent assumptions, and practical context for students and science learners.

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