Class 11 Physics · NCERT notes for NEET

Motion in a Straight Line: NCERT Notes, Formulas and Tricks for NEET

Kinematics in one dimension: displacement, velocity, acceleration, the equations of motion, graphs, relative velocity and free fall. It is the language of all of mechanics, and NEET usually asks at least one direct question from it, often through a graph.

• 1. Position, displacement, speed and velocity
• 2. Acceleration and equations of motion
• 3. Motion graphs (x-t, v-t, a-t)
• 4. Relative velocity in one dimension
• 5. Free fall and vertical motion
• Quick revision

Free chapter: the full notes are open for everyone.

Open the interactive version (simulations and tap-to-answer quick checks).

Topic 1

Position, displacement, speed and velocity

Walk 1 km to the shop and come back home: your speedometer app says 2 km, but your displacement is exactly zero.

• Motion along a straight line is rectilinear motion. Choose an origin O and a positive direction; position $x$ then carries a sign.
• Rest and motion are relative: a body moves if its position changes with time with respect to the chosen frame of reference.
• Path length (distance) is the total length actually travelled. It is a scalar and is never negative.
• Displacement $\Delta x=x_2-x_1$ is the change in position. It is a vector in 1D: it can be positive, negative or zero.
• $|\Delta x|\le$ path length. They are equal only if the body moves in one direction without turning back.
• Average velocity $=\Delta x/\Delta t$; average speed $=$ path length / time. So average speed $\ge$ |average velocity|.
• Instantaneous velocity $v=dx/dt$ is the slope of the tangent to the $x$-$t$ graph. Instantaneous speed is simply $|v|$.
• Uniform motion: equal displacements in equal time intervals. Then instantaneous velocity equals average velocity at every instant.
• SI unit ${\text{m s}}^{-1}$. Convert with $1{\text{km h}}^{-1}=\frac{5}{18}{\text{m s}}^{-1}$ (so $72{\text{km h}}^{-1}=20{\text{m s}}^{-1}$).
• Round trip back to the start: displacement and average velocity are zero, but average speed is not.

Formulas

• $\bar{v}=\frac{x_2-x_1}{t_2-t_1}=\frac{\Delta x}{\Delta t}$
  Average velocity; sign shows direction
• $\text{average speed}=\frac{\text{total path length}}{\text{total time}}$
  Always positive
• $v={\lim }_{\Delta t\to 0}\frac{\Delta x}{\Delta t}=\frac{dx}{dt}$
  Instantaneous velocity
• $v_{\text{avg}}=\frac{2v_1{v}_2}{v_1+v_2}$
  Equal distances covered at $v_1$ and $v_2$ (harmonic mean)
• $v_{\text{avg}}=\frac{v_1+v_2}{2}$
  Equal time intervals at $v_1$ and $v_2$ (arithmetic mean)

Tips, tricks and hacks

Common mistakes

NEET traps

NEET gives a body on a circle or a round trip and asks average velocity: the answer is zero (or chord/time), not path/time.

"Instantaneous speed equals magnitude of instantaneous velocity" is TRUE, even though average speed and |average velocity| can differ.

Displacement can be zero while distance is not; but distance can never be zero while displacement is non-zero.

Topic 2

Acceleration and equations of motion

Double your bike's speed and you need four times the road to stop. That one fact is why speed limits near schools are so low.

• Average acceleration $\bar{a}=\Delta v/\Delta t$; instantaneous acceleration $a=dv/dt=d^{2}x/d{t}^2$. SI unit ${\text{m s}}^{-2}$.
• Acceleration can be positive, negative or zero. Its sign alone does not tell you whether the body speeds up.
• The body speeds up when $v$ and $a$ have the same sign and slows down when they have opposite signs.
• $v$ can be zero while $a$ is not zero, for example at the top of a vertical throw.
• For constant $a$ only: $v=u+at$, $s=ut+\frac{1}{2}a{t}^2$, $v^2=u^2+2as$, and $\bar{v}=(u+v)/2$.
• These equations can be derived from the area under the $v$-$t$ graph (rectangle $ut$ + triangle $\frac{1}{2}a{t}^2$) or by integration.
• Stopping distance $d_s=v_0^2/(2a)$ for deceleration of size $a$: $d_s\propto v_0^2$.
• Reaction time: NCERT drops a ruler between your fingers; if it falls $d$ before you catch it, $t_r=\sqrt{2d/g}$ (21 cm gives about 0.2 s).
• Non-uniform acceleration: use calculus. $v=\int adt$, $x=\int vdt$, or $vdv=adx$ when $a$ depends on $x$.

Formulas

• $v=u+at$
  Velocity after time $t$ (constant $a$)
• $s=ut+\frac{1}{2}a{t}^2$
  Displacement in time $t$
• $v^2=u^2+2as$
  Use when time is not given or not asked
• $s=\frac{u+v}{2}t$
  Use when $a$ is not given
• $s_n=u+\frac{a}{2}(2n-1)$
  Displacement in the $n$th second
• $d_s=\frac{v_0^2}{2a}$
  Stopping distance; $a$ = size of deceleration
• $a=v\frac{dv}{dx}$
  Use when $a$ is given as a function of $x$

Tips, tricks and hacks

Common mistakes

NEET traps

"A body can have zero velocity and non-zero acceleration": TRUE (top of a throw). "Non-zero velocity and zero acceleration": also TRUE (uniform motion).

Stopping-distance questions often give the speed in km/h; convert first or use the ratio $d_s\propto v^2$ directly.

The ratio of distances in the $n$th and $(n+1)$th second from rest is $(2n-1):(2n+1)$, not $n^2:(n+1{)}^2$.

Topic 3

Motion graphs (x-t, v-t, a-t)

One look at a graph's slope and area gives velocity, acceleration and displacement without a single equation; NEET loves to hide the answer in a graph.

• Slope of the $x$-$t$ graph is velocity: chord slope gives average velocity, tangent slope gives instantaneous velocity.
• At rest: $x$-$t$ is a horizontal line. Uniform motion: a straight inclined line (positive slope for $v>0$, negative for $v<0$).
• Uniform acceleration: $x$-$t$ is a parabola. Curving upward means $a>0$, curving downward means $a<0$.
• Slope of the $v$-$t$ graph is acceleration. A straight $v$-$t$ line means constant acceleration.
• Area under the $v$-$t$ graph is displacement; area below the time axis counts as negative. Adding all areas as positive gives distance.
• Area under the $a$-$t$ graph is the change in velocity $\Delta v$.
• Where the $v$-$t$ graph crosses the time axis, the body turns back. At that point the $x$-$t$ graph has a maximum or minimum.
• Impossible graphs: an $x$-$t$ curve with two values of $x$ at one $t$, a speed-time graph going below zero, or a vertical $x$-$t$ segment (infinite velocity).
• Area under the $x$-$t$ graph has no useful meaning; never use it.

Formulas

• $v=\frac{dx}{dt}$ = slope of $x$-$t$
  Tangent slope at that instant
• $a=\frac{dv}{dt}$ = slope of $v$-$t$
  Acceleration from the velocity-time graph
• $\Delta x={\int }_{t_1}^{t_2}vdt$
  Signed area under $v$-$t$ = displacement
• $\Delta v={\int }_{t_1}^{t_2}adt$
  Area under $a$-$t$ = change in velocity

Tips, tricks and hacks

Common mistakes

NEET traps

NEET shows an $x$-$t$ parabola and asks the sign of $a$: decide from curvature (∪ or ∩), not from whether $x$ is rising.

A question asks "which graph is not possible": look for two $x$ values at one $t$, negative speed or negative distance.

For $v$-$t$ graphs, the answer option with "total area" is distance and "net area" is displacement; both are usually offered.

Topic 4

Relative velocity in one dimension

From a train at 80 km/h, a car on the next road at 80 km/h looks parked; a train coming the other way flashes past at 160 km/h.

• Velocity of A relative to B is how fast A's position changes as seen by B: $v_{AB}=v_A-v_B$.
• $v_{BA}=v_B-v_A=-v_{AB}$: same size, opposite sign.
• Same direction: relative speed is the difference of speeds. Opposite directions: relative speed is the sum.
• If $v_A=v_B$, the gap stays constant and their $x$-$t$ graphs are parallel lines.
• If A is behind B but faster, the gap shrinks at $v_A-v_B$; the $x$-$t$ lines meet at the overtaking point.
• Relative position $x_{AB}=x_A-x_B$ and relative acceleration $a_{AB}=a_A-a_B$ follow the same rule.
• Two freely falling bodies have the same acceleration $g$, so $a_{AB}=0$ and their relative velocity stays constant.
• Time to meet or overtake = initial gap / relative speed (constant velocities).

Formulas

• $v_{AB}=v_A-v_B$
  Velocity of A relative to B; use signs with one common positive direction
• $v_{BA}=-v_{AB}$
  Reverse the observer, reverse the sign
• $t=\frac{d}{|v_{AB}|}$
  Time to close a gap $d$ at constant relative speed
• $t=\frac{L_1+L_2}{|v_{AB}|}$
  Time for two trains of lengths $L_1$, $L_2$ to cross each other completely

Tips, tricks and hacks

Common mistakes

NEET traps

"Two balls in free fall: their relative velocity is zero" is FALSE in general; it is constant, and zero only if they had equal velocities at the start.

Answer options often swap $v_{AB}$ and $v_{BA}$; the sizes are equal, so check the sign or direction asked.

If two bodies move with equal velocities, their $x$-$t$ graphs are parallel; parallel $v$-$t$ graphs mean equal accelerations, not equal velocities.

Topic 5

Free fall and vertical motion

In a vacuum tube a feather and a coin hit the bottom together; on the Moon, astronaut David Scott showed the same with a hammer and a feather.

• Free fall: motion under gravity alone, air resistance neglected. Acceleration is $g\approx 9.8{\text{m s}}^{-2}$ downward, the same for all masses.
• With upward as positive, $a=-g$ for the whole flight: going up, at the top and coming down.
• Dropped from rest: $v=gt$, $h=\frac{1}{2}g{t}^2$, $v=\sqrt{2gh}$, $t=\sqrt{2h/g}$.
• Thrown up with speed $u$: time to top $u/g$, maximum height $H=u^2/(2g)$, total time back to the hand $2u/g$.
• At the highest point $v=0$ but $a=g$ downward; the body is not in equilibrium.
• Without air resistance, time up = time down, and the body passes every height with the same speed going up and coming down.
• Galileo's law of odd numbers: from rest, distances in successive equal time intervals are $1:3:5:7\dots$; total distances are $1:4:9:16$.
• For a ball thrown up: $y$-$t$ is a downward parabola, $v$-$t$ a straight line of slope $-g$ crossing zero at the top, $a$-$t$ a flat line at $-g$.
• NCERT example: thrown up at $20{\text{m s}}^{-1}$ from a 25 m building ($g=10$): it rises 20 m more and hits the ground after 5 s.

Formulas

• $H=\frac{u^2}{2g}$
  Maximum height for a vertical throw with speed $u$
• $T=\frac{2u}{g}$
  Total time of flight back to the launch point
• $v=\sqrt{2gh}$
  Speed after falling height $h$ from rest
• $t=\sqrt{\frac{2h}{g}}$
  Time to fall height $h$ from rest
• $y=y_0+ut-\frac{1}{2}g{t}^2$
  Height with upward positive; works for the whole flight

Tips, tricks and hacks

Common mistakes

NEET traps

"At the top of its path, velocity and acceleration are both zero": FALSE; only velocity is zero.

"Distances in successive seconds are in ratio 1 : 4 : 9": FALSE; that is total distance. Successive seconds give 1 : 3 : 5.

A ball thrown down with speed $u$ and one thrown up with the same $u$ from the same height hit the ground with the same speed (but at different times).

Quick revision: Motion in a Straight Line the night before NEET

1. Distance is never negative; displacement can be $+$, $-$ or zero; $|\Delta x|\le$ distance.
2. Average speed $\ge$ |average velocity|; equal only with no turning back.
3. Equal distances at $v_1$, $v_2$: $v_{avg}=2v_1{v}_2/(v_1+v_2)$; equal times: $(v_1+v_2)/2$.
4. $1{\text{km h}}^{-1}=5/18{\text{m s}}^{-1}$; 72 km/h = 20 m/s.
5. $v=u+at$, $s=ut+\frac{1}{2}a{t}^2$, $v^2=u^2+2as$ need constant $a$.
6. $s_n=u+\frac{a}{2}(2n-1)$; from rest, successive-second distances go $1:3:5$.
7. Stopping distance $\propto v_0^2$: double the speed, four times the distance.
8. Speeding up if $v$ and $a$ have the same sign; slowing down if opposite.
9. Slope of $x$-$t$ = velocity; slope of $v$-$t$ = acceleration.
10. Area under $v$-$t$ = displacement (signed) or distance (all positive); area under $a$-$t$ = $\Delta v$.
11. $v$-$t$ crossing zero = turning point; $x$-$t$ ∪ means $a>0$, ∩ means $a<0$.
12. $v_{AB}=v_A-v_B$, $v_{BA}=-v_{AB}$; opposite directions add speeds.
13. Two freely falling bodies: relative velocity constant.
14. Vertical throw: $H=u^2/2g$, $T=2u/g$, time up = time down.
15. At the top: $v=0$ but $a=g$ downward.

Practise Motion in a Straight Line questions

Topic-wise practice from Level 1 to Level 3, NCERT-based questions and NEET PYQs, with a full solution after every answer.