Laws of Motion formulas
Class 11 physics formula sheet for NEET and JEE: the key equations of NCERT chapter 4, the special cases questions are built on, and diagrams where they help.
By Sreeraj P, M.Sc Physics · 10+ years teaching NEET and JEE
Most used formulasOther formulas and cases
1Newton's laws and momentum
- 1st law: inertia (of rest, motion, direction); mass measures inertia. Coin tossed in a train: falls behind if train accelerates, ahead if it retards, back into hand at constant velocity.
- 3rd law: action and reaction act on different bodies, never cancel; not valid for pseudo forces.
Impulse = area under $F$–$t$ graph. Catching with hands lowered (more time) → smaller force. 1 kg wt $=9.8$ N; 1 N $=10^5$ dyne.
| Change in momentum | $|\Delta\vec p|$ |
|---|---|
| Hits wall and stops / rebounds with same speed | $mv$ / $2mv$ (normal to wall) |
| Rebounds with $v_2$ after hitting at $v_1$ | $m(v_1+v_2)$ |
| Strikes at angle $\theta$ with wall, same speed | $2mv\sin\theta$ ($2mv\cos\theta$ if $\theta$ from normal) |
| Uniform circular motion through angle $\theta$ | $2mv\sin\frac\theta2$ (towards centre) |
| Projectile: launch → top / launch → landing | $mu\sin\theta$ / $2mu\sin\theta$ (downward) |
| Variable mass / stream | Force |
|---|---|
| Machine gun, $n$ bullets of mass $m$ in time $t$ | $F=\dfrac{nmv}t$ |
| Bullets holding a plate up: stop / bounce back | $\dfrac{nmv}t=Mg$ / $\dfrac{2nmv}t=Mg$ |
| Liquid jet (area $A$, density $\rho$) on wall: stops / rebounds | $\rho Av^2$ / $2\rho Av^2$; rebounds at $v'$: $\rho Av(v+v')$; at angle $\theta$ to wall: $2\rho Av^2\sin\theta$ |
| Flow through a 90° pipe bend | $\sqrt2\rho Av^2$ |
| Sand dropped on conveyor belt moving at $u$ | $F=u\dfrac{dm}{dt}$ |
| Falling chain on table ($y$ fallen) | force on table $=\dfrac{3Mgy}L$ |
| Rocket | thrust $=u\dfrac{dm}{dt}$; $v=u\ln\dfrac{m_0}{m}-gt$ |
Recoil: $V=\dfrac{mv}M$. Bullet embeds in block: $V=\dfrac{mv}{M+m}$. Man walks $s$ on a boat: boat moves $\dfrac{ms}{M+m}$. Shell at rest breaks into two: $\dfrac{v_1}{v_2}=\dfrac{m_2}{m_1}$; into three: third momentum balances the other two. Shell exploding at top of projectile path: use $Mu\cos\theta=m_1v_1+m_2v_2$ (horizontal); one piece retraces path → other gets $\dfrac{(M+m_1)u\cos\theta}{m_2}$.
2Equilibrium, tension and pulleys
Bob held at angle $\theta$ by a horizontal force. Chain hanging between walls at angle $\theta$: end tension $\dfrac{mg}{2\sin\theta}$, middle $\dfrac{mg}2\cot\theta$. Heavy rope (mass $m$) holding $M$: tension at distance $x$ from bottom $\left(M+\dfrac{mx}L\right)g$. Straightening a rope with a weight needs infinite tension.
Blocks pulled on a smooth surface (same for contact forces when pushed). Vertical chain pulled up by $F$: $T=\dfrac{m_{\text{below}}}{m_{\text{total}}}F$.
| System (smooth, light pulley) | $a$ | $T$ |
|---|---|---|
| Atwood ($m_1>m_2$) | $\dfrac{(m_1-m_2)g}{m_1+m_2}$ | $\dfrac{2m_1m_2g}{m_1+m_2}$; thrust on pulley $2T$ |
| $m_1$ on table, $m_2$ hanging | $\dfrac{m_2g}{m_1+m_2}$ | $\dfrac{m_1m_2g}{m_1+m_2}$ |
| … with friction $\mu$ on table | $\dfrac{(m_2-\mu m_1)g}{m_1+m_2}$ | $\dfrac{m_1m_2g(1+\mu)}{m_1+m_2}$ |
| $m_1$ on incline $\theta$, $m_2$ hanging | $\dfrac{(m_2-m_1\sin\theta)g}{m_1+m_2}$ | $\dfrac{m_1m_2g(1+\sin\theta)}{m_1+m_2}$ |
| Double incline $\alpha$, $\beta$ | $\dfrac{(m_2\sin\beta-m_1\sin\alpha)g}{m_1+m_2}$ | $\dfrac{m_1m_2g(\sin\alpha+\sin\beta)}{m_1+m_2}$ |
| Three blocks $M_1$, $M_2$ (table), $M_3$ | $\dfrac{(M_1-M_3)g}{M_1+M_2+M_3}$ | — |
| Pulley accelerating up at $a_0$ | replace $g$ by $g+a_0$ | |
- Constraints: total string length constant. Movable pulley holding $m_2$, $m_1$ on the other end: $a_1=2a_2$; hanging $m_2$: $a_2=\dfrac{(m_2-2m_1)g}{m_2+4m_1}$; $m_1$ on a smooth table: $a_2=\dfrac{m_2g}{4m_1+m_2}$. Force $F$ on a light pulley whose string pulls a block: $T=F/2$, block $a=\dfrac F{2m}$, pulley $\dfrac F{4m}$.
- Rope climbing: $T=m(g+a)$. Man of mass $M$ in a box $m$ pulling a rope over a pulley to hold it still: scale reads $\dfrac{(M-m)g}2$.
3 more sections and 14 formulas in the full chapter
- 3Frames and apparent weight2 formulas · 1 case table
- 4Friction5 formulas · 3 case tables
- 5Circular motion dynamics7 formulas · 1 diagram
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