Q.Derive the work-energy theorem for a variable force. OR Write relation between kinetic energy and momentum and with the help of this relation, answer, if the momentum of a lighter and heavy body are same, then K.E. of which body will be greater ?
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The Work-Energy Theorem: From Intuition to Precision
Imagine pushing a heavy box across the floor. The harder you push and the farther it slides, the faster it moves when you let go. That connection — the push (force) over a distance (displacement) changing the box's speed — is exactly what the Work-Energy Theorem captures.
The Intuition First
Think of work as the "currency" that buys motion. When you do work on an object, you transfer energy to it. That energy shows up as kinetic energy — the energy of motion. The more work you do, the more the object's kinetic energy changes.
If you push a stationary ball, it starts moving. If you push a moving ball in the same direction, it speeds up. If you push against its motion, it slows down. In every case, the work done equals the change in the ball's kinetic energy.
Work is done by a force on an object. The object's kinetic energy changes by exactly that amount (assuming no other forces do work).
The Precise Statement
Wnet=ΔK=Kf−Ki
Where:
- Wnet is the net work done on the object (the total work from all forces combined)
- Kf is the final kinetic energy
- Ki is the initial kinetic energy
And kinetic energy is defined as:
K=21mv2
So the theorem can also be written as:
Wnet=21mvf2−21mvi2
Why "Net" Work Matters
This is the most common point of confusion. The theorem uses net work — the work done by the net force (the vector sum of all forces). If you push a box and friction opposes it, the net work is the work you do minus the work friction does. Only that net amount changes the kinetic energy.
If you push a box at constant speed, your work is positive, but friction does equal negative work. The net work is zero, so kinetic energy doesn't change — the box keeps moving at the same speed. Your work didn't "disappear"; it was dissipated as heat by friction.
A Simple Derivation (for constant force)
Consider a constant net force Fnet acting on an object of mass m over a displacement s. From Newton's second law:
Fnet=ma
From kinematics (constant acceleration):
vf2=vi2+2as
Multiply both sides by 21m:
21mvf2=21mvi2+mas
But mas=Fnets=Wnet, so:
21mvf2=21mvi2+Wnet
Rearranging:
Wnet=21mvf2−21mvi2=ΔK
The theorem holds even for variable forces and curved paths — the derivation uses calculus then, but the result is the same.
What It Tells You (and What It Doesn't) …
The work-energy theorem for a variable force is derived by expressing the small work done over an infinitesimal displacement and integrating. …
Even when the force varies with position, breaking the motion into infinitesimal steps and integrating shows that total work done still equals the change in kinetic energy.
Consider a variable force F(x) acting on a body of mass m moving along the x-axis. Over an infinitesimally small displacement dx, the force can be treated as constant, so the small work done is:
dW=Fdx
By Newton's second law, F=mdtdv. Using the chain rule dtdv=dxdvdtdx=vdxdv:
dW=mvdxdvdx=mvdv
Integrating as the body's speed changes from v1 to v2:
W=∫v1v2mvdv=21mv22−21mv12
…
Showing the 12 most recent of 16 on this concept.
- CBSE 2026Set ANNUAL1 markMCQQ.Match the column - select the correct definition (from Column B) for the term 'Energy' (Column A):(a) change in linear momentum(b) motion opposing force(c) loss of energy(d) rate of change of momentum(e) ability of doing work(f) rate of doing work
›Reveal solutionSolution
Energy is defined as the capacity of a body to do work.
In mechanics, the energy possessed by a body (whether kinetic, due to its motion, or potential, due to its position/configuration) is precisely defined as its capacity to perform work — a body with more energy can do more work before that capacity is exhausted. Thi …
- CBSE 2026Set ANNUAL1 markMCQQ.A particle moves under the effect of a force F = αx from x = 0 to x = d. The work done in the process is (α = constant)(a) αd(b) αd^2(c) (1/2)αd^2(d) zero
›Reveal solutionSolution
W = integral of alpha x dx from 0 to d = ½ alpha d^2. Answer (C).
For a position-dependent force, work done = integral of F dx.
…
- CBSE 2025Set ANNUAL1 markMCQQ.A force of 40 N acts on body of mass 5 kg which is initially at rest. What is the amount of work done in the first 10 s?(a) 1600 J(b) -1600 J(c) 400 J(d) -400 J
›Reveal solutionSolution
Work done by a constant force on a body starting from rest is W=F2t2/(2m). With the numbers as printed the arithmetic gives 16{,}000 J (ten times option (a)); using m=50 kg — the value consistent with the printed options, and a very plausible OCR/typo of '5 kg' for '50 kg' in the source paper — gives exactly 1600 J, so that is the answer selected here.
Acceleration: a=F/m
Distance covered from rest in time t: s=(1/2)at2
Work done by the applied force: W=F×s=F×(1/2)(F/m)t2=F2t2/(2m)
With F=40 N, t=10 s, and m=50 kg (see note above):
a=40/50=0.8 m/s²
s=(1/2)(0.8)(10)2=40 m
W=40×40=1600 J
…
- CBSE 2024Set ANNUAL1 markMCQQ.The dimensional formula of work done is the same as the dimensional formula of(a) Momentum(b) Power(c) Energy(d) Torque
›Reveal solutionSolution
Work and energy are dimensionally identical because work done equals the change in energy (work-energy theorem).
Work done, W = Force x displacement = [MLT^-2][L] = [M L^2 T^-2].
Energy (kinetic or potential) has the same dimensional formula [M L^2 T^-2], since energy is literally defined as the capacity to do work.
Check the other options:
- Momentum = mass x velocity = [M][LT^-1] = [MLT^-1] (different).
- Power = work/time = [ML^2T^-3] (different). …
- CBSE 2024Set ANNUAL1 markMCQQ.Energy is(a) capacity of doing work(b) rate of doing work(c) change of work(d) none of these
›Reveal solutionSolution
Energy is defined as the capacity of a body or system to do work.
By definition, energy is the ability of a system to perform work; a body with more energy can do more work. This is why work and energy share the same SI unit, the joule, and the same dimensional fo …
- CBSE 2024Set ANNUAL1 markMCQQ.Area under force-displacement curve represents(a) Velocity(b) Acceleration(c) Impulse(d) Work done
›Reveal solutionSolution
Work done = force x displacement (for a varying force, the integral of F dx), which is exactly the area enclosed under an F-x graph.
For a constant force, W = F x s, which is literally the area of the rectangle under a horizontal F-x line. For a varying force, W = ∫ F dx, the area under the (possibly curved) F-x graph, found by summi …
- CBSE 2024Set ANNUAL1 markMCQQ.Which of the following is a unit of energy?(a) Horse power(b) Joule second(c) Kilowatt hour(d) Watt.
›Reveal solutionSolution
Kilowatt hour = power × time = energy, unlike horse power and watt (power) or joule-second (action).
…
- CBSE 2024Set ANNUAL1 markQ.State whether true or false: Work is a vector quantity.
›Reveal solutionSolution
False. Work W = F.d (dot product of force and displacement) is a scalar quantity.
Work done by a force F causing a displacement d is defined as W = F . d = |F||d| cos(theta), the scalar (dot) product of the two vectors. …
- CBSE 2023Set ANNUAL1 markMCQQ.The work done by a body against friction always results in(1) loss of kinetic energy(2) loss of potential energy(3) gain of kinetic energy(4) gain of potential energy
›Reveal solutionSolution
Friction is dissipative: work done against it always drains kinetic energy from the system, converting it irreversibly into heat.
Friction opposes relative motion, so when a body moves against friction, the friction force does negative work on the body. By the work-energy theorem, negative work done on a body reduces its kinetic energy. This lost mechanical energy is not stored as potential energy (friction is not a conservative force, so no potential energy can be associated with it) - it is dissipated as heat at the su …
- CBSE 2023Set ANNUAL1 markMCQQ.A raindrop of mass 1 g falling from a height of 1 km hits the ground with a speed of 50 ms^-1. If the resistive force is proportional to the speed of the drop, then the work done by the resistive force is (Take g = 10 ms^-2)(1) 10 J(2) -10 J(3) 8.75 J(4) -8.75 J
›Reveal solutionSolution
Apply the work-energy theorem: total work done (by gravity and the resistive force together) equals the change in kinetic energy; solve for the unknown resistive-force work.
Given:
Mass, m = 1 g = 0.001 kg
Height fallen, h = 1 km = 1000 m
Final speed, v = 50 m/s
g = 10 m/s^2
Work done BY gravity (positive, since displacement is in the direction of gravity):
W_gravity = mgh = 0.001 x 10 x 1000 = 10 J
Change in kinetic energy (drop starts from rest):
ΔKE = (1/2)mv^2 - 0 = (1/2)(0.001)(50)^2 = (1/2)(0.001)(2500) = 1.25 J
…
- CBSE 2023Set ANNUAL1 markQ.Mention two physical quantities which have same dimensions as that of work.
›Reveal solutionSolution
Work has dimensional formula [ML2T−2]; energy and torque share exactly this formula.
Work is defined as W=F⋅d, giving dimensions [MLT−2][L]=[ML2T−2]. By the work-energy theorem, energy (kinetic energy 21mv2, potential energy mgh, heat, etc.) has exactly the same dimensional formula, since energy is measured in the same unit, the joule. Torque is defined as τ=F×r (force times perpendicular distance), which also gives [MLT−2][L]=[ML2T−2] — the same d …
- CBSE 2023Set ANNUAL1 markMCQQ.Which of the following quantities represents the change in kinetic energy of any object?(a) Force(b) Mass(c) Linear momentum(d) Work
›Reveal solutionSolution
The work-energy theorem directly identifies work as the physical quantity equal to the change in kinetic energy of a body.
Starting from Newton's second law, F = ma = m(dv/dt). The work done by this force over a displacement ds is:
dW = F ds = m (dv/dt) ds = m v dv (since ds/dt = v)
Integrating from initial speed u to final speed v:
W = integral of m v dv from u to v = (1/2) m v^2 - (1/2) m u^2 = KE_final - KE_initial = Delta KE
…
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