JEE Mains Physics · Motion in a Straight Line

Average Speed, Average Velocity and Relative Velocity

Average speed is total distance over total time, never the mean of the speeds; relative velocity is one velocity minus the other, so speeds add for bodies moving towards each other and subtract for bodies moving the same way.

All Motion in a Straight Line notes JEE Mains Physics strategy

Why this matters

Fourteen PYQs, eleven of them multiple choice, and one from 2026. Nine ask for an average speed over a journey in two or three legs, and five for a relative velocity between trains, cars or a walker on an escalator. Each one is a single division once the total distance and the total time are written down.

Concept 1 of 2: Average speed over a journey in legs

Average speed is the total distance divided by the total time. When the legs are equal DISTANCES, the slow leg takes longer, so the average sits below the plain mean of the speeds: it is the harmonic mean. When the legs are equal TIMES, each speed counts equally and the plain mean is right. For anything else, find the time of each leg and add.

Definition

Average speed $= \dfrac{\text{total distance}}{\text{total time}}$ ; average velocity $= \dfrac{\text{displacement}}{\text{time}}$ .
Two equal distances at $v_1, v_2$ : $\bar v = \dfrac{2v_1v_2}{v_1 + v_2}$ , so $\dfrac{2}{\bar v} = \dfrac{1}{v_1} + \dfrac{1}{v_2}$ .
Three equal distances: $\bar v = \dfrac{3}{1/v_1 + 1/v_2 + 1/v_3}$ .
Two equal times at $v_1, v_2$ : $\bar v = \dfrac{v_1 + v_2}{2}$ .
Half the distance at $v_1$ , the other half in two equal times at $v_2$ and $v_3$ : the second half's speed is $\dfrac{v_2 + v_3}{2}$ ; then take the harmonic mean with $v_1$ .
Unequal legs, such as x and then 3x/2: add the leg times $x/v_1 + 1.5x/v_2$ and divide the total distance by them.
Speeding up uniformly from rest to v covers the distance it would at v/2.

Average speed

\bar v = \frac{\text{total distance}}{\text{total time}} \qquad \text{equal distances: } \bar v = \frac{2v_1v_2}{v_1 + v_2}

Worked example

A car covers the first third of a road at

20\ \text{km/h}

, the next third at

30\ \text{km/h}

and the last third at

60\ \text{km/h}

. Find its average speed.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2024 · 9 April 2024 · Q2Moderate

Example 1 · Motion in a Straight Line · Average Speed, Average Velocity and Relative Velocity

A particle moving in a straight line covers half the distance with speed

6\text{ }m/s

. The other half is covered in two equal time intervals with speeds 9

m/s

and

15\text{ }m/s

respectively. The average speed of the particle during the motion is:

Averaging two speeds over equal distances

Equal distances take unequal times, so the plain mean is wrong. At 3 km/h and 5 km/h over equal distances the average is 2·3·5/8 = 3.75 km/h, not 4 km/h.

Average velocity on a round trip is zero

If the body comes back to its start, the displacement is zero, so the average velocity is zero whatever the speeds. The average speed is not zero.

Equal times inside an equal-distance leg

When half the distance is split into two equal TIMES, average those two speeds plainly first. Only then take the harmonic mean with the first half.

Drill 8 more on average speed over a journey in legs

Concept 2 of 2: Relative velocity in one dimension

The velocity of B as seen from A is B's velocity minus A's. Pick one direction as positive and give every velocity its sign. Two trains moving towards each other then close the gap at the sum of their speeds; two moving the same way close it at the difference. To cross, a train must cover the total length to be passed at that relative speed.

Definition

$v_{BA} = v_B - v_A$ , with signs. Opposite directions: the speeds add. Same direction: they subtract.
The ground seen from B moves at $-v_B$ .
Crossing time $= \dfrac{\text{sum of the lengths to pass}}{\text{relative speed}}$ : two trains pass each other in $(L_1 + L_2)/v_{rel}$ ; a train clears a tunnel or bridge in $(L_{train} + L_{tunnel})/v$ .
A passenger watching another train go by sees only that train's length pass.
Walking on a moving escalator: the speeds add, so $\dfrac{1}{t} = \dfrac{1}{t_1} + \dfrac{1}{t_2}$ , giving $t = \dfrac{t_1t_2}{t_1 + t_2}$ .
Convert km/h to m/s with $\times \dfrac{5}{18}$ : 36 km/h is 10 m/s, 72 km/h is 20 m/s.

Relative velocity

v_{BA} = v_B - v_A \qquad t_{cross} = \frac{L_1 + L_2}{v_{rel}}

Worked example

A train 150 m long running at

72\ \text{km/h}

overtakes a train 100 m long running at

36\ \text{km/h}

on a parallel track. How long does the overtaking take?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2023 · 13 April 2023 · Q96Moderate

Example 2 · Motion in a Straight Line · Average Speed, Average Velocity and Relative Velocity

A passenger sitting in a train A moving at

90\text{ }km/h

observes another train B moving in the opposite direction for

8\text{ }s

. If the velocity of the train

B

54\text{ }km/h

, then length of train B is:

Forgetting the train's own length

A train has cleared a tunnel only when its last coach leaves, so the distance is tunnel length plus train length. Using the tunnel alone gives too short a time.

Subtracting speeds for trains approaching each other

With one direction taken positive, the other train's velocity is negative. v_B − v_A then has the size of the SUM of the speeds.

Leaving speeds in km/h

Lengths are in metres and times in seconds, so convert first. 108 km/h is 30 m/s; 18 km/h is 5 m/s.

Drill 4 more on relative velocity in one dimension

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Average speed over a journey in legs
Average speed
$\bar v = \frac{\text{total distance}}{\text{total time}} \qquad \text{equal distances: } \bar v = \frac{2v_1v_2}{v_1 + v_2}$
Relative velocity in one dimension
Relative velocity
$v_{BA} = v_B - v_A \qquad t_{cross} = \frac{L_1 + L_2}{v_{rel}}$

Watch out for (6)

Averaging two speeds over equal distances
→ Average speed over a journey in legs
Average velocity on a round trip is zero
→ Average speed over a journey in legs
Equal times inside an equal-distance leg
→ Average speed over a journey in legs
Forgetting the train's own length
→ Relative velocity in one dimension
Subtracting speeds for trains approaching each other
→ Relative velocity in one dimension
Leaving speeds in km/h
→ Relative velocity in one dimension

Test yourself on Motion in a Straight Line

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