Time-speed-distance

Time–Speed–Distance – Study Notes (IIT‑JEE / Competitive Exam Level)

Scope – All questions that involve time, speed (or velocity) and distance, including relative‑motion, average speed, piecewise motion, unit conversion and graphical interpretation.
These notes are written for the Haryana Police Sub‑Inspector (SI) Quantitative Aptitude exam but are equally useful for IIT‑JEE and other competitive tests.

1. Fundamental Relations

Basic Formulae

[ \boxed{v=\frac{d}{t}}, \qquad \boxed{d=v,t}, \qquad \boxed{t=\frac{d}{v}} ]

Note:
Speed is a scalar; velocity is a vector. For most quantitative aptitude problems we use speed (magnitude only).

2. Unit Conversion – The “Speed‑to‑Time” Trick

Why?
(1 \text{ km} = 1000 \text{ m}) and (1 \text{ h} = 3600 \text{ s}).

Quick Formula

[ v_{\text{m/s}} = v_{\text{km/h}}\times \frac{1000}{3600} = v_{\text{km/h}}\times\frac{5}{18} ]

3. Average Speed vs. Average Velocity

• Average Speed
  [ v_{\text{avg}} = \frac{\text{Total Distance}}{\text{Total Time}} ] (scalar, no sign)

• Average Velocity
  [ \vec{v}_{\text{avg}} = \frac{\text{Displacement}}{\text{Total Time}} ] (vector, direction matters)

Typical Pitfall – In “round‑trip” problems students often mistakenly divide the total distance by total time when the displacement is zero.
Always check whether the problem asks for speed (use total distance) or velocity (use displacement).

4. Relative Speed

Derivation (Two objects towards each other)

Let (d) be the initial separation.
Time to meet (t) satisfies

[ d = (v_1+v_2)t ;;\Longrightarrow;; t = \frac{d}{v_1+v_2} ]

5. Piecewise Motion – “Average Speed” of a Journey

If an object travels several segments with different speeds:

[ t_{\text{total}} = \sum_i \frac{d_i}{v_i}, \qquad v_{\text{avg}} = \frac{\sum_i d_i}{t_{\text{total}}} ]

Tricky Point – Do not average the speeds directly; average the distances against the times.

6. Graphical Interpretation

• **Distance–