NDA Physics · Work, Energy and Power

Energy — Kinetic, Potential, and Conservation

Energy is the capacity to do work. The two mechanical forms are kinetic energy (½mv², due to motion) and potential energy (mgh, due to position). On a frictionless system the total stays constant — energy only changes form, never disappears.

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Why this matters

This is the largest subtopic in the chapter — 10 PYQs from 2017 to 2026, spanning EASY recall (what is potential energy, the conservation statement) to MODERATE numericals (find the landing speed from the drop height) and the chapter's two HARD outliers (potential energy from a force law, and kinetic-energy change across reference frames). Master ½mv², mgh, and the PE-to-KE conversion of a falling body and you cover the bulk of the chapter's marks.

Concept 1 of 5

Kinetic energy — energy of motion (½mv²)

Intuition

Anything that moves can do work on something else when it stops — a moving hammer drives a nail, moving water turns a turbine. That stored capacity due to motion is kinetic energy, and it grows with the SQUARE of the speed, so doubling the speed quadruples the kinetic energy.

Definition

Kinetic energy is the energy a body has by virtue of its motion: $KE = \tfrac{1}{2}mv^2$ . It is a scalar, measured in joules, and is always positive or zero. Because it depends on $v^2$ , the kinetic energy rises steeply with speed.

Kinetic energy

KE = \tfrac{1}{2}mv^2

KEkinetic energy (J)
mmass of the body (kg)
vspeed of the body (m/s)

Worked example

A 4 kg ball moves at 3 m/s. Find its kinetic energy.

Use $KE = \tfrac{1}{2}mv^2$ .
Substitute $m = 4$ kg, $v = 3$ m/s.
$KE = \tfrac{1}{2}(4)(3^2) = \tfrac{1}{2}(4)(9) = 18$ J.

Answer:

KE = 18

Practice this conceptself-check · 4 quick reps

Try it yourself

An object of mass 2 kg has 100 J of kinetic energy. Find its speed.

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
A 1 kg body moves at 4 m/s. Find its kinetic energy.
2.
If the speed of a body doubles, its kinetic energy becomes how many times the original?
3.
A 2 kg object has KE 36 J. Find its speed.
4.
Can kinetic energy be negative?

From the bank · past-year question

Example 1Work, Energy and PowerEASY

An object of mass 2000 g possesses 100 J kinetic energy. The object must be moving with a speed of

[Q141 · Sep · 2021]

Kinetic energy grows with the SQUARE of speed

Doubling the speed multiplies kinetic energy by 4, tripling it by 9. A body at twice the speed of another (same mass) carries four times the kinetic energy — not twice.

Convert grams to kilograms before substituting

The mass in

\tfrac{1}{2}mv^2

must be in kilograms. A mass given as 2000 g is 2 kg; forgetting to convert gives an answer 1000 times too large.

Concept 2 of 5

Potential energy — energy of position (mgh)

Intuition

Lift a body up and you store energy in it — release it and gravity converts that stored energy back into motion. This energy of position or configuration is potential energy; for a body raised to a height it equals the work done against gravity to put it there.

Definition

Potential energy is the energy a body possesses by virtue of its position or shape (configuration). For a body of mass $m$ raised through a height $h$ near the Earth's surface, the gravitational potential energy is $PE = mgh$ . It is measured from a chosen reference level (usually the ground), where $PE = 0$ .

Gravitational potential energy

PE = mgh

PEgravitational potential energy (J)
mmass of the body (kg)
gacceleration due to gravity ( $\approx 9.8$ m/s²)
hheight above the reference level (m)

Worked example

A 3 kg block rests on a shelf 2 m above the floor. Find its gravitational potential energy relative to the floor. Take

g = 10

m/s².

Use $PE = mgh$ .
Substitute $m = 3$ kg, $g = 10$ m/s², $h = 2$ m.
$PE = 3 \times 10 \times 2 = 60$ J.

Answer:

PE = 60

Practice this conceptself-check · 4 quick reps

Try it yourself

How high must a 2 kg ball be raised to store 400 J of gravitational potential energy? Take

g = 10

m/s².

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Energy possessed by a body due to its position or shape is called what?
2.
A 5 kg body is at height 3 m. Find its PE. (g = 10)
3.
At what height is the gravitational PE usually taken as zero?
4.
A stretched spring stores which kind of energy?

From the bank · past-year question

Example 2Work, Energy and PowerEASY

The energy possessed by a body due to its change in position or shape is called

[Q82 · Apr · 2022]

Potential energy is about POSITION or SHAPE — not motion

A body raised to a height, a compressed spring, or a stretched bow all store potential energy because of their configuration. Energy due to motion is kinetic energy — keep the two definitions distinct.

Concept 3 of 5

Conservation of energy — PE converts to KE as a body falls

Intuition

Drop a ball: at the top it is all potential energy and no motion; at the bottom it is all kinetic energy and no height. Energy is not lost — it simply changes form. On a frictionless path the sum PE + KE is the same at every point.

Definition

The law of conservation of energy states that energy can be neither created nor destroyed — only transformed from one form to another; the total energy of an isolated system is constant. For a body falling freely (no friction), mechanical energy is conserved: $PE + KE = \text{constant}$ . So a body dropped from rest through height $h$ arrives with $\tfrac{1}{2}mv^2 = mgh$ , giving $v = \sqrt{2gh}$ — and at that instant its kinetic energy equals the potential energy it started with.

Energy conservation for a freely falling body

mgh = \tfrac{1}{2}mv^2 \;\Rightarrow\; v = \sqrt{2gh}

mghpotential energy at the top (J)
\tfrac{1}{2}mv^2kinetic energy at the bottom (J)
vlanding speed (m/s)

As the ball descends, potential energy mgh converts into kinetic energy ½mv². On a frictionless track the total mechanical energy never changes.

Worked example

A ball of mass 0.32 kg is released from a height where it has 625 J of potential energy. With what speed does it hit the ground (ignore air resistance)?

All the potential energy converts to kinetic energy: $\tfrac{1}{2}mv^2 = 625$ J.
$\tfrac{1}{2}(0.32)v^2 = 625 \Rightarrow v^2 = 625 / 0.16 = 3906.25$ .
$v = \sqrt{3906.25} = 62.5$ m/s.

Answer:

v = 62.5

m/s.

Practice this conceptself-check · 4 quick reps

Try it yourself

A 2 kg body is dropped from a balloon at rest 50 m above the ground. Find its total mechanical energy when dropped, and its speed just before landing. Take

g = 9.8

m/s².

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
State the law of conservation of energy in one line.
2.
A body falls freely from height h from rest. Its landing speed?
3.
A ball thrown up reaches highest point B from A. Compare KE at A with PE at B.
4.
For which kind of system is total energy always conserved — open, closed, or isolated?

From the bank · past-year question

Example 3Work, Energy and PowerMODERATE

A ball of mass 320 g has 625 J potential energy when released from a height. The speed with which it will hit the ground is

[Q97 · Sep · 2024]

Energy is conserved for an ISOLATED system

The total energy stays constant only when no energy crosses the system boundary — that is the isolated case. For real bodies, friction and air resistance carry mechanical energy away as heat and sound, so MECHANICAL energy alone is not conserved, but total energy still is.

At the bottom of a free fall, KE equals the starting PE

For a body dropped from rest, the kinetic energy on landing equals the potential energy it had at the top:

\tfrac{1}{2}mv^2 = mgh

. Set them equal — do not add them.

Drill 4 more on conservation of energy — pe converts to ke as a body falls

Concept 4 of 5

Conservative forces and energy transformations

Intuition

Some forces store energy you can get back (lift a weight, recover it by lowering) — these are conservative. Others dissipate it irreversibly as heat or sound (friction, air drag) — these are non-conservative. The NDA tests this as a recall fact ("which is NOT a conservative force?") and as a sequence-of-transformations question (an apple falling).

Definition

A conservative force does work that is path-independent and recoverable (gravity, spring force, electrostatic force). A non-conservative (dissipative) force does path-dependent work that turns mechanical energy into heat or sound (friction, air resistance, viscous drag). Energy continually transforms between forms; the table below lists the facts and the canonical falling-apple sequence the bank tests.

Item	Classification / sequence	Note
Gravitational force	Conservative	work depends only on height change
Spring (elastic) force	Conservative	energy fully recovered on release
Electrostatic force	Conservative	path-independent work
Frictional force	Non-conservative	dissipates energy as heat — the bank's answer "Which is NOT a conservative force?" — the answer is friction.
Air resistance / drag	Non-conservative	removes mechanical energy as heat
Apple falling to ground	GPE → KE → Sound → Heat	PE turns to motion, then a thud, then heat on impact The correct transfer sequence: gravitational PE → kinetic → sound → heat.

Friction is the standard "not conservative" answer; the falling-apple sequence runs gravitational PE → KE → sound → heat.

Practice this conceptself-check · 4 quick reps

Try it yourself

Give the correct sequence of energy transformations when an apple falls from a tree and lands on the ground.

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Which of these is NOT a conservative force: gravity, spring force, electrostatic force, friction?
2.
Name one conservative force.
3.
When an apple hits the ground, KE converts mainly into which two forms?
4.
Is air resistance conservative or non-conservative?

From the bank · past-year question

Example 4Work, Energy and PowerEASY

Which one of the following is not a conservative force?

[Q149 · Sep · 2021]

Friction is the standard NON-conservative force

Gravity, spring force, and electrostatic force are conservative (recoverable, path-independent). Friction and air resistance are non-conservative — they turn mechanical energy into heat and cannot give it back.

The falling-apple sequence ends in HEAT, not sound

The order is gravitational PE → kinetic → sound → heat. KE turns to sound (the thud) AND heat on impact — distractors reorder these or put heat before kinetic energy.

Drill 1 more on conservative forces and energy transformations

Concept 5 of 5

Kinetic energy and its change depend on the reference frame

Intuition

Speed is measured relative to an observer, and kinetic energy depends on speed — so two observers moving differently disagree on a body's kinetic energy AND on how much it changes. This is the chapter's hardest idea: even the CHANGE in kinetic energy is frame-dependent, because the work done by a force depends on the displacement seen in each frame.

Definition

Because $KE = \tfrac{1}{2}mv^2$ uses the speed relative to the observer, kinetic energy is not absolute — it differs from frame to frame. The change $\Delta K$ over a process is likewise frame-dependent: for a body that starts at speed $u$ (in a moving frame) and gains an extra $\sqrt{2gh}$ by falling, $\Delta K = mgh + mu\sqrt{2gh}$ — larger than the rest-frame value $mgh$ by the cross term $mu\sqrt{2gh}$ . The deeper reason: work $= F \times$ displacement, and the displacement of the ground differs between the frames.

Frame-dependent change in kinetic energy

\Delta K_{S'} = mgh + mu\sqrt{2gh} \;>\; \Delta K_{S} = mgh

\Delta K_SKE change in the rest frame = mgh
\Delta K_{S'}KE change in a frame moving with speed u
urelative speed of the two frames (m/s)

Worked example

A ball is dropped from rest through height

h

. In the ground frame, find the change in its kinetic energy. Then state qualitatively how the change compares in a frame moving upward at speed

u

Ground frame: starts at rest, lands at $v = \sqrt{2gh}$ , so $\Delta K_S = \tfrac{1}{2}m(2gh) = mgh$ .
Moving frame $S'$ : the ball already has downward speed $u$ at release and reaches $u + \sqrt{2gh}$ .
$\Delta K_{S'} = \tfrac{1}{2}m[(u+\sqrt{2gh})^2 - u^2] = mgh + mu\sqrt{2gh}$ , which is larger by the positive term $mu\sqrt{2gh}$ .

Answer:

\Delta K_S = mgh

; the change is LARGER in the moving frame

S'

Practice this conceptself-check · 4 quick reps

Try it yourself

Two observers — one standing still, one in a train moving at constant velocity — measure the kinetic energy of the same rolling ball. Do they get the same value? Why?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Is kinetic energy the same in every reference frame?
2.
A ball dropped from rest falls height h. Change in KE in the ground frame?
3.
In a frame moving relative to the ground, is the CHANGE in a falling body's KE generally the same as in the ground frame?
4.
Why does the change in KE differ between frames?

From the bank · past-year question

Example 5Work, Energy and PowerHARD

A ball is dropped from rest from height

h

in frame

S

. Frame

S'

moves upwards with speed

u

. In

S'

, ball has initial downward speed

v

. Pertaining to the change in kinetic energy (

\Delta K

) from release to just before hitting the ground as measured in

S

and

S'

separately, which one is correct?

[Q58 · Apr · 2026]

Even the CHANGE in kinetic energy is frame-dependent

It is tempting to think

\Delta K

is the same for all observers since both see the same fall. It is not — the moving frame adds a cross term

mu\sqrt{2gh}

, because work depends on the displacement seen in that frame.

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 (4)

Kinetic energy — energy of motion (½mv²)
Kinetic energy
$KE = \tfrac{1}{2}mv^2$
Potential energy — energy of position (mgh)
Gravitational potential energy
$PE = mgh$
Conservation of energy — PE converts to KE as a body falls
Energy conservation for a freely falling body
$mgh = \tfrac{1}{2}mv^2 \;\Rightarrow\; v = \sqrt{2gh}$
Kinetic energy and its change depend on the reference frame
Frame-dependent change in kinetic energy
$\Delta K_{S'} = mgh + mu\sqrt{2gh} \;>\; \Delta K_{S} = mgh$

Reference tables (1)

Conservative forces and energy transformations6 rows

Item	Classification / sequence	Note
Gravitational force	Conservative	work depends only on height change
Spring (elastic) force	Conservative	energy fully recovered on release
Electrostatic force	Conservative	path-independent work
Frictional force	Non-conservative	dissipates energy as heat — the bank's answer "Which is NOT a conservative force?" — the answer is friction.
Air resistance / drag	Non-conservative	removes mechanical energy as heat
Apple falling to ground	GPE → KE → Sound → Heat	PE turns to motion, then a thud, then heat on impact The correct transfer sequence: gravitational PE → kinetic → sound → heat.

Friction is the standard "not conservative" answer; the falling-apple sequence runs gravitational PE → KE → sound → heat.

Watch out for (8)

Kinetic energy grows with the SQUARE of speed
→ Kinetic energy — energy of motion (½mv²)
Convert grams to kilograms before substituting
→ Kinetic energy — energy of motion (½mv²)
Potential energy is about POSITION or SHAPE — not motion
→ Potential energy — energy of position (mgh)
Energy is conserved for an ISOLATED system
→ Conservation of energy — PE converts to KE as a body falls
At the bottom of a free fall, KE equals the starting PE
→ Conservation of energy — PE converts to KE as a body falls
Friction is the standard NON-conservative force
→ Conservative forces and energy transformations
The falling-apple sequence ends in HEAT, not sound
→ Conservative forces and energy transformations
Even the CHANGE in kinetic energy is frame-dependent
→ Kinetic energy and its change depend on the reference frame

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1Work, Energy and PowerMODERATE

A rigid body of mass 2 kg is dropped from a stationary balloon kept at a height of 50 m from the ground. The speed of the body when it just touches the ground and the total energy when it is dropped from the balloon are respectively (acceleration due to gravity = 9·8 m/s

^2

)

[Q97 · Sep · 2019]

Example 2Work, Energy and PowerMODERATE

The correct sequence of energy transfer that occurs when an apple falls to the ground is

[Q59 · Apr · 2019]

Example 3Work, Energy and PowerEASY

Suppose, a ball of mass M is thrown upwards from a point A and it reaches up to the highest point B and returns back to point A. Which one among the following is correct ?

[Q75 · Sep · 2025]

Example 4Work, Energy and PowerEASY

The energy is always conserved for a system which is

[Q74 · Sep · 2025]

Example 5Work, Energy and PowerEASY

Which one of the following statements about energy is correct?

[Q112 · Sep · 2017]

Drill every past-year question on this subtopic

10 questions from the bank — paginated, with cart and Word-export support.

Drill the 10 questions