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Physics · Class 11

Slope & signed area

Motion in a Straight Line

Differentiating a position graph gives velocity; differentiating velocity gives acceleration. Integrating the signed velocity area gives displacement, while integrating acceleration gives the change in velocity. Initial position and initial velocity supply the constants needed to reconstruct the journey.

Open the interactive experiment →

Move between position, velocity and acceleration.

நிலை வரைபடத்தின் சாய்வு திசைவேகம்; திசைவேக வரைபடத்தின் சாய்வு முடுக்கம். திசைவேக வரைபடத்தின் குறியுடன் கூடிய பரப்பளவு இடப்பெயர்ச்சி. முடுக்கத்தின் பரப்பளவு திசைவேக மாற்றம்.

Key notes, relationships and model limits

Key relationship

v = dx/dt; a = dv/dt; Δx = ∫v dt; Δv = ∫a dt

Remember

Differentiating a position graph gives velocity; differentiating velocity gives acceleration. Integrating the signed velocity area gives displacement, while integrating acceleration gives the change in velocity. Initial position and initial velocity supply the constants needed to reconstruct the journey.

Common trap

The area below the time axis subtracts from displacement; total distance requires the area of |v|.

Model limits

a(t) = a₀ + jt; v(t) = u + a₀t + ½jt²; x(t) = x₀ + ut + ½a₀t² + ⅙jt³. Readouts use exact integrals; shaded polygons are sampled illustrations.

What you will learn

Connect slopes and signed areas across three graphs, including the required initial values.

Try the experiment

Scrub the common time marker. Watch the two shaded areas grow and compare them with position and velocity changes.

Think of it this way

A running total and the rate at which it changes are two connected views of the same process.

Worked example

With u = 4 m/s and a = −1 m/s², from 0 to 8 s the positive velocity area is +8 m and negative area is −8 m. Displacement is 0 m; distance is 16 m.

Why this matters

Slope and area methods also work when acceleration changes and constant-acceleration shortcuts fail.

Before you start

Signed numbers · coordinates · gradients · area under a graph

Check your understanding

What does the signed area under an acceleration–time graph represent?

Read the explanation

∫a dt = vfinal − vinitial. Its units are (m/s²) × s = m/s.

Course connection: CBSE XI Chapter 2; NEET and JEE Main Unit 2; JEE Advanced Mechanics. Check the selected syllabus for relative-velocity scope.