Velocity-Time Graph Area (Distance) – Tutorial

Distance from Velocity-Time Graph — Detailed Guide

This calculator computes distance traveled (or more precisely, displacement) using the area under the velocity-time graph. It also separately calculates total distance when negative velocities are present (using |v|).

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1) Core concept: What does "area" represent on this graph?

1.1 Relationship between velocity (v) and position (x)

In physics, velocity is the rate of change of position:

v(t) = dx/dt

Reversing this:

x(b) - x(a) = ∫_a^b v(t) dt

So the area under the v–t graph gives the displacement over the time interval.

1.2 Are "displacement" and "total distance" the same?

Not always.

• Displacement: difference between final and initial position (can be signed).
• Total distance: total path traveled — always non-negative.

If velocity is never negative (always moving forward), they're equal. If velocity goes negative (moving backward), displacement decreases. That's why this calculator gives two results:

• Displacement (signed area) = ∫ v(t) dt
• Total distance (absolute area) = ∫ |v(t)| dt

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2) What is the Trapezoidal Rule? (Numerical integration)

In practice, we rarely know v(t) as an exact formula. We have measured data points:

(t₀, v₀), (t₁, v₁), ...

The trapezoidal rule treats the graph as a straight line segment in each interval and computes the area of each trapezoid.

2.1 Area for a single interval

Between (t1, v1) and (t2, v2):

Area ≈ (v1 + v2)/2 × (t2 - t1)

• (t2 - t1) = Δt (time interval)
• (v1 + v2)/2 = average velocity in this interval (linear assumption)

2.2 Total over all intervals

Total ≈ Σ (v_i + v_i+1)/2 × (t_i+1 - t_i)

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3) Units: m/s vs km/h?

• m/s: Area result is in meters (multiplying by seconds)
• km/h: Converted to m/s first: 1 km/h = 1000/3600 m/s
• km/s: Converted to m/s: 1 km/s = 1000 m/s

Results are shown in both meters and kilometers.

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4) What happens with negative velocity?

Negative velocity means moving in the opposite direction. For example, going forward then backward:

• Displacement can be small (or even 0)
• Total distance can be large (because you actually traveled both ways)

So the results show two separate values:

• Displacement: signed area (area below x-axis is negative)
• Total distance: absolute area (negative parts counted as positive)

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5) Per-interval table

For each pair of adjacent points, the calculator shows:

• Δt
• Average velocity
• Segment displacement
• Segment total distance (absolute)

This makes it clear exactly how much each interval contributes.

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6) Quick example

Points:

• (0, 0)
• (2, 4)

Δt = 2, avg v = (0+4)/2 = 2 → area = 2×2 = 4. If unit is m/s, distance = 4 meters.

Note: This calculator is for educational use. With noisy measurement data, more frequent sampling gives better results.