Motion in a Straight Line— Notes, Formulas & Revision

x-t graph plots position against time. Slope at any point = instantaneous velocity.

A horizontal line means rest (v = 0). A straight inclined line means uniform velocity.

A curved line (parabola) means constant acceleration — x(t) = x₀ + v₀t + ½at².

Concave-up parabola → a > 0; concave-down → a < 0.

Two objects meet when their x-t curves intersect.

v-t graph: slope = acceleration; area under the curve = displacement.

For uniform acceleration, v-t is a straight line: v(t) = v₀ + at.

A v-t line crossing zero means the object reversed direction.

Positive area = forward displacement; negative area = backward displacement.

a-t graph slope = jerk (da/dt). Area under a-t = change in velocity.

A step function in a-t means acceleration jumps suddenly between phases.

If a = 0, velocity is constant; if a is constant, velocity changes linearly.

Uniform motion: velocity is constant; acceleration is zero.

x-t graph is a straight line; v-t is horizontal; a-t is zero.

Distance = speed × time. Displacement = velocity × time.

Uniform acceleration: a is constant in magnitude and direction.

Three kinematic equations: v = v₀ + at, x = v₀t + ½at², v² = v₀² + 2ax.

The v-t graph is a straight line; x-t is a parabola.

Free fall is motion under gravity alone (ignoring air resistance).

Acceleration = g = 9.8 m/s² downward (≈10 m/s² for quick estimates).

From rest at height h: fall time t = √(2h/g); impact speed v = √(2gh).

Thrown up with speed u: max height = u²/(2g); time up = u/g; total flight = 2u/g.

Velocity of A relative to B: v_AB = v_A − v_B.

If both move in same direction, relative velocity = difference.

If in opposite directions, relative velocity = sum.

Ground frame and moving frames give different values for the same motion.

Two objects meet when their positions are equal: x_A(t) = x_B(t).

For uniform motion: t_meet = (x₀_B − x₀_A)/(v_A − v_B).

If v_A = v_B, they never meet (unless starting at same point).

If v_A − v_B and x₀_B − x₀_A have opposite signs, they never meet.

Reading motion from graphs is a core JEE skill — always check slope and curvature.

On an x-t graph: horizontal = rest; straight up = uniform v; curve = acceleration.

On a v-t graph: horizontal = uniform v; line through zero = reversal; parabola = variable a.

When a = f(t), we integrate: v(t) = v₀ + ∫a(t') dt', x(t) = x₀ + ∫v(t') dt'.

Common cases: a = kt gives parabolic v(t); a = sin(ωt) gives oscillatory v.

Numerical integration (Euler, Runge–Kutta) is used when no closed form exists.