Work, Energy, Power and Collisions — NEET Physics

Energy methods are often the fastest route through a mechanics problem: instead of tracking forces at every instant, you compare energy at the start and end. This chapter builds work (for constant and variable forces), the work–energy theorem, potential energy and its link to force, conservation of mechanical energy, power, and the momentum-and-energy accounting of collisions — closing with the vertical-circle results that combine energy with circular motion. Master these and a large, recurring slice of NEET Physics becomes bookkeeping.

1. Work done by a constant force

$W=\vec{F}\cdot \vec{d}=Fd\cos \theta$

Work is the component of force along the displacement, times the distance.

• A 10 N force pulling at 60° over 5 m does $W=10\times 5\times \cos 60°=25$ J.
• Positive work when $\theta <90°$ (force aids motion), negative when $\theta >90°$ (friction, braking), and zero at $\theta =90°$.
• A force perpendicular to motion does no work — the normal force on a level slide, the tension on a mass in circular motion, and gravity on horizontal motion all do zero work.

Unit: joule (J) $=$ N·m.

2. Work done by a variable force

When the force changes with position, work is the area under the force–displacement graph:

$W=\int Fdx$

The key case is a spring, where $F=kx$ grows linearly with extension. The work to stretch it to $x$ is the triangular area:

$W_{\text{spring}}=\frac{1}{2}k{x}^2$

3. The work–energy theorem

The net work done on a body equals the change in its kinetic energy:

$W_{\text{net}}=\Delta KE=\frac{1}{2}m{v}^2-\frac{1}{2}m{u}^2$

• 100 J of net work on a 2 kg body from rest gives $\frac{1}{2}(2)v^2=100\Rightarrow v=10$ m/s.

This is a shortcut that skips the force–time details — whenever a question asks only for a speed given work, energy, or a height, reach for this instead of $F=ma$.

A useful identity: since $p=mv$, kinetic energy is $KE=\frac{p^2}{2m}$ — so at fixed mass, doubling the momentum quadruples the kinetic energy.

4. Potential energy and the force link

Potential energy is stored energy of configuration:

$U_{\text{grav}}=mgh,U_{\text{spring}}=\frac{1}{2}k{x}^2$

For any conservative force, force is the negative gradient of potential energy:

$F=-\frac{dU}{dx}$

so the force points "downhill" in energy — toward lower $U$. This is why a stretched spring pulls back and a raised mass falls.

5. Conservation of mechanical energy

When only conservative forces act (gravity, springs — no friction), mechanical energy is constant:

$KE+PE=\text{constant}$

• A body dropped from height $h$ arrives with $\frac{1}{2}m{v}^2=mgh\Rightarrow v=\sqrt{2gh}$. From 20 m ($g=10$): $v=\sqrt{400}=20$ m/s — independent of mass.
• A pendulum trades PE at the top for KE at the bottom and back, forever (ideally).

If friction acts, mechanical energy is not conserved — the lost energy appears as heat, and you must include the work done against friction.

6. Power

$P=\frac{W}{t}=\vec{F}\cdot \vec{v}=Fv$

• Average power is total work over total time; instantaneous power is $Fv$ at that speed.
• A pump raising 100 kg of water 10 m in 20 s ($g=10$) delivers $P=\frac{100\times 10\times 10}{20}=500$ W.
• SI unit watt (W); $1\text{hp}\approx 746$ W.

7. Collisions

Momentum is conserved in every collision. Kinetic energy is conserved only in an elastic one.

The coefficient of restitution compares relative speeds after and before:

$e=\frac{v_2-v_1}{u_1-u_2}$

One-dimensional elastic collision — the standard results:

• Equal masses, one at rest: they exchange velocities — the incoming ball stops, the target moves off at the original speed (Newton's cradle).
• Very heavy target at rest: the light ball bounces straight back at nearly the same speed (like a ball off a wall).
• Perfectly inelastic, equal masses: they stick and move at $v/2$.

Worked example 7.1. A 2 kg body at 6 m/s strikes a stationary 4 kg body and they stick together. Common speed? Momentum: $2\times 6=(2+4)V\Rightarrow V=12/6=2$ m/s. KE falls from 36 J to $\frac{1}{2}(6)(2^2)=12$ J — 24 J lost as heat/deformation, as expected for a perfectly inelastic collision.

8. The vertical circle — energy meets circular motion

For a body looping a vertical circle of radius $r$ on a string, the minimum speed at the top is where gravity alone provides the centripetal force ($mg=m{v}^2/r$):

$v_{\text{top}}=\sqrt{gr}$

Using energy conservation from top to bottom (a drop of $2r$):

$v_{\text{bottom}}=\sqrt{v_{\text{top}}^2+4gr}=\sqrt{gr+4gr}=\sqrt{5gr}$

So to just complete the loop, the body needs $\sqrt{5gr}$ at the bottom. This combination of energy conservation and centripetal condition is a NEET staple.

9. Common traps NEET sets here

• Dropping the $\cos \theta$ in work — a perpendicular force does zero work.
• Assuming KE is conserved in every collision — only elastic ones; momentum is always conserved.
• Using energy conservation with friction present — then mechanical energy is not conserved.
• Forgetting $KE=p^2/2m$ — doubling momentum quadruples KE, not doubles it.
• Confusing the equal-mass elastic result — velocities exchange; the incoming ball stops, it does not continue.
• Mis-stating the vertical-circle speeds — $\sqrt{gr}$ at the top, $\sqrt{5gr}$ at the bottom.

10. Memory aids

• "Perpendicular = no work" — normal force, circular tension, horizontal gravity.
• "Momentum always, KE only if elastic" — the one-line collision rule.
• "Exchange on equal, bounce on heavy" — the two elastic special cases.
• "√(gr) top, √(5gr) bottom" — vertical-circle minimum speeds.
• "KE = p²/2m" — the momentum–energy bridge.

11. Exam protocol

1. Use $W=Fd\cos \theta$; a perpendicular force does no work.
2. For springs and variable forces, work is the area under the F–x graph ($\frac{1}{2}k{x}^2$).
3. Reach for the work–energy theorem when you need only speeds.
4. Apply conservation of mechanical energy when friction is absent ($v=\sqrt{2gh}$ for a drop).
5. Use $F=-dU/dx$ to link force and potential energy.
6. Compute power as $W/t$ or $Fv$.
7. In collisions, conserve momentum always, KE only if elastic; use the equal-mass exchange and $v/2$ results.
8. For a vertical loop, remember $\sqrt{gr}$ (top) and $\sqrt{5gr}$ (bottom).