Motion in a Straight Line

2.1 Core concepts

The focus here is rectilinear motion — motion of objects along a straight line — and within that, mostly motion with constant acceleration (NCERT §2.1, p. 13). The objects are treated as point objects, an approximation valid when the size of the object is much smaller than the distance it moves. Kinematics itself is defined as the study of ways to describe motion without going into its causes; the causes (forces) come in the next chapter on laws of motion (NCERT §2.1, p. 13).

The first analytical concept is instantaneous velocity. The velocity v at an instant t is defined as the limit of the average velocity as the time interval Δt approaches zero: v = lim(Δt→0) Δx/Δt = dx/dt, i.e., the differential coefficient of position with respect to time (NCERT §2.2, p. 14, Eq. 2.1a–b). Graphically, the instantaneous velocity is the slope of the tangent to the position-time curve at that instant; this is illustrated through Fig. 2.1 (p. 14), where successive chords P₁P₂ over shrinking Δt converge to a tangent at t = 4 s, and the numerical values in Table 2.1 (Δt from 2.0 s down to 0.010 s) converge to 3.84 m s⁻¹ — an explicit demonstration of the limit definition (NCERT §2.2, p. 14). For uniform motion, the velocity is the same as the average velocity at all instants (NCERT §2.2, p. 15). Instantaneous speed (or just speed) is the magnitude of instantaneous velocity; velocities of +24.0 m s⁻¹ and −24.0 m s⁻¹ both have speed 24.0 m s⁻¹ (NCERT §2.2, p. 15). An important asymmetry: although average speed over a finite interval is greater than or equal to the magnitude of average velocity, instantaneous speed equals the magnitude of instantaneous velocity at every instant (NCERT §2.2, p. 15).

The same limit-building exercise yields acceleration. Average acceleration a over an interval is the change of velocity divided by the time interval: a = (v₂ − v₁)/(t₂ − t₁) = Δv/Δt, with SI unit m s⁻² (NCERT §2.3, p. 15, Eq. 2.2). On a v–t plot, average acceleration is the slope of the straight line connecting (v₁, t₁) and (v₂, t₂). Instantaneous acceleration a = lim(Δt→0) Δv/Δt = dv/dt is the slope of the tangent to the v–t curve at that instant (NCERT §2.3, p. 16, Eq. 2.3). Because velocity is a vector with magnitude and direction, acceleration can arise from a change in speed, in direction, or in both; it can be positive, negative, or zero (NCERT §2.3, p. 16). On a position–time graph the curve bends upward for positive acceleration, downward for negative acceleration, and is a straight line for zero acceleration (NCERT §2.3, p. 16, Fig. 2.2). NCERT also states the geometric identity that will reappear throughout mechanics: the area under the v–t curve between t₁ and t₂ equals the displacement of the object over that interval (NCERT §2.3, p. 16; Summary §5, p. 22).

The core result is the trio of kinematic equations for uniformly accelerated motion: v = v₀ + at (Eq. 2.4), x = v₀t + ½at² (Eq. 2.6), and v² = v₀² + 2ax (Eq. 2.8). These connect five quantities — initial velocity v₀, final velocity v, acceleration a, time t and displacement x — so that any three determine the other two (NCERT §2.4, pp. 17–18, Eqs. 2.9a). If the starting position at t = 0 is x₀ rather than zero, x is simply replaced by (x − x₀) (NCERT §2.4, p. 18, Eqs. 2.9b–c). The equations follow from the average-velocity identity v̄ = (v + v₀)/2 (valid only for constant acceleration) and can also be derived by calculus by integrating a = dv/dt and v = dx/dt. The calculus method has the advantage that it generalises to non-uniform acceleration via a = v(dv/dx) (NCERT §2.4, Example 2.2, p. 18). NCERT also derives the same equations graphically from the area under a v–t plot: for uniform velocity u, the rectangular area is uT (Fig. 2.4); for uniformly accelerated motion, the trapezium area = ½(v − v₀)t + v₀t = v₀t + ½at² (Fig. 2.5), which is exactly Eq. 2.6 (NCERT §2.4, p. 17).

Three classical applications cement the equations. Free fall near the Earth's surface (air resistance neglected) is a special case of motion with uniform acceleration; the magnitude of g ≈ 9.8 m s⁻² is directed downward (NCERT §2.4, Example 2.4, p. 19). Galileo's law of odd numbers states that the distances traversed by a body falling from rest during equal successive time intervals are in the ratio 1 : 3 : 5 : 7 : 9 : 11 …; the cumulative distances follow 1 : 4 : 9 : 16 : 25 … (NCERT §2.4, Example 2.5, p. 20, Table 2.2). The stopping distance of a vehicle under uniform deceleration is dₛ = −v₀²/(2a) — proportional to the square of the initial velocity, so doubling speed quadruples stopping distance, a result of great relevance to road safety (NCERT §2.4, Example 2.6, pp. 20–21). Reaction time is the time a person takes to observe, think and act in response to a situation; it can be measured very simply by the ruler-drop experiment, in which a freely-falling ruler caught between thumb and forefinger gives tᵣ from d = ½g·tᵣ² (NCERT §2.4, Example 2.7, p. 21).

NCERT closes with a careful Summary and a "Points to Ponder" list emphasising that quantities in kinematic equations are algebraic (can be ±) and depend on the chosen positive direction of the axis; that the equations are valid only for constant acceleration whereas the definitions of v and a are always exact; that the sign of acceleration does not by itself say whether speed is increasing or decreasing (that depends on the relative direction of a and v); and that a particle can be momentarily at rest while still accelerating — e.g., at the highest point of a vertical throw v = 0 but a = −g (NCERT Points to Ponder, p. 23). These subtleties are favourite CUET trap territory.

2.2 Definitions to memorise

2.3 Diagrams / processes to remember

Eight high-yield figures are worth knowing. Fig. 2.1 (p. 14) plots the position–time curve x = 0.08 t³ and shows how successive chords P₁P₂ over shrinking Δt converge to the tangent at t = 4 s; this is the visual of the limit definition of velocity. Table 2.1 (p. 14) tabulates the corresponding numerical values of Δx/Δt for Δt ranging from 2.0 s down to 0.010 s centred at t = 4 s; the sequence converges to 3.84 m s⁻¹ — students should be ready for an MCQ that asks them to "identify the instantaneous velocity from the table." Fig. 2.2 (p. 16) sketches three position–time curves: one curving upward (positive acceleration), one downward (negative acceleration), and one straight line (zero acceleration). Fig. 2.3 (p. 16) shows four characteristic velocity–time graphs of constant acceleration, including the diagnostic case (d) in which an object reverses direction at t = t₁ under negative acceleration. Fig. 2.4 (p. 17) demonstrates that for uniform velocity u, the rectangular area uT under the v–t graph equals the displacement uT — the area-equals-displacement theorem in its simplest form. Fig. 2.5 (p. 17) generalises this to uniform acceleration, where the trapezium area = v₀t + ½at², thereby visually deriving Eq. 2.6.

Fig. 2.7 (pp. 19–20) carries the free-fall triple: (a) a vs t is a horizontal line at −g, (b) v vs t is a straight line of slope −g, (c) y vs t is a downward parabola. CUET often tests students on identifying which of three plots correctly shows free fall — this triple is the answer key. Fig. 2.8 (p. 21) sketches the ruler-drop experiment for reaction time: a ruler held between thumb and forefinger is released, and the distance fallen d before being caught gives the reaction time via tᵣ = √(2d/g). Table 2.2 (p. 20) tabulates distances during successive equal intervals in free fall to verify Galileo's odd-number ratio 1 : 3 : 5 : 7 : 9 : 11 …; the cumulative distances follow 1 : 4 : 9 : 16 : 25, a square-number ratio that is a classic distractor.

The two process skills to internalise are (i) graph reading: slope of x–t at a point = instantaneous velocity; slope of v–t at a point = instantaneous acceleration; area under v–t = displacement; area under a–t = change in velocity; and (ii) kinematic-equation selection: identify which of v₀, v, a, t, x are known and which is unknown, then pick the kinematic equation that excludes the un-asked variable.

2.4 Common confusions / NTA trap points

• The sign of acceleration does not tell you whether speed is increasing or decreasing; that depends on whether a is along or opposite to v (NCERT Points to Ponder 2 & 3, p. 23).
• A particle can be momentarily at rest with non-zero acceleration — e.g., a ball at the highest point of its upward throw has v = 0 but a = −g (NCERT Points to Ponder 4, p. 23).
• Average speed ≥ |average velocity| over a finite interval, but instantaneous speed = |instantaneous velocity| at every instant (NCERT §2.2, p. 15; Exercise 2.11, p. 25).
• The kinematic equations (2.9) are valid only for motion with constant magnitude and direction of acceleration; definitions of v and a (Eqs. 2.1 and 2.3) are always exact (NCERT Points to Ponder 6, p. 23).
• Quantities in kinematic equations are algebraic (can be ±); always assign signs based on the chosen positive direction of the axis before substituting (NCERT Points to Ponder 1 & 5, p. 23).
• The ratio 1 : 3 : 5 : 7 … is for distances in successive equal intervals of free fall; the ratio 1 : 4 : 9 : 16 … is for cumulative distances at the end of each interval. Mixing the two is a textbook NTA distractor (NCERT Example 2.5, p. 20).
• Doubling speed quadruples stopping distance, not double — because dₛ ∝ v₀² (NCERT Example 2.6, p. 21).
• The slope of a chord on the x–t graph gives average velocity, not instantaneous; instantaneous needs the tangent (NCERT Fig. 2.1, p. 14).
• Path length is always ≥ |displacement|, with equality only for unidirectional motion; an object that returns to its starting point has zero displacement but non-zero path length.
• Free fall in NCERT means motion under gravity with air resistance neglected — students should not import drag corrections without being asked.
• For a body thrown vertically up, the time of ascent equals the time of descent only if the launch and landing heights are the same; with a building/cliff offset, this no longer holds (NCERT Example 2.3, pp. 18–19).
• The area-under-v–t = displacement rule treats signed area — area below the t-axis is negative displacement, which matters whenever the v–t curve crosses the axis (NCERT §2.3, p. 16, Fig. 2.3d).

2.5 Key formulas