CUET UG Physics Mechanics Work Energy Power Conservation Laws Collisions

Must‑Know

• Work done by a constant force is ( W = \vec{F} \cdot \vec{s} = Fs\cos\theta ); for example, when a force of 10 N displaces an object by 5 m at 60°, ( W = 10 \times 5 \times \cos60^\circ = 25 \text{ J} ).
• Work done is zero when displacement is perpendicular to force; e.g., centripetal force does no work in uniform circular motion.
• Kinetic energy ( K = \frac{1}{2}mv^2 ); a 2 kg object moving at 3 m/s has ( K = \frac{1}{2} \times 2 \times 9 = 9 \text{ J} ).
• Work-Energy Theorem: ( W_{\text{net}} = \Delta K ); if a net work of 50 J is done on a body, its kinetic energy increases by 50 J.
• Conservative forces conserve mechanical energy; e.g., gravitational force is conservative, friction is non-conservative.
• Potential energy for gravity near Earth’s surface: ( U = mgh ); a 1 kg mass at 10 m height has ( U = 1 \times 9.8 \times 10 = 98 \text{ J} ).
• For a spring, ( U = \frac{1}{2}kx^2 ); a spring of ( k = 200 \text{ N/m} ) compressed by 0.1 m stores ( U = \frac{1}{2} \times 200 \times (0.1)^2 = 1 \text{ J} ).
• Power ( P = \frac{W}{t} = \vec{F} \cdot \vec{v} ); a force of 10 N acting on a body moving at 4 m/s delivers ( P = 10 \times 4 = 40 \text{ W} ).
• 1 horsepower = 746 W; used in vehicle engine ratings.
• In elastic collisions, both momentum and kinetic energy are conserved; e.g., two identical balls colliding head-on exchange velocities.
• In inelastic collisions, momentum is conserved but kinetic energy is not; e.g., a bullet embedding in a block.
• Perfectly inelastic collision: maximum kinetic energy loss; objects stick together after collision.
• Coefficient of restitution ( e = \frac{v_2 - v_1}{u_1 - u_2} ); for perfectly elastic collision, ( e = 1 ); for perfectly inelastic, ( e = 0 ).
• In one-dimensional elastic collision between two masses ( m_1 ) and ( m_2 ), final velocities are:
  ( v_1 = \frac{(m_1 - m_2)u_1 + 2m_2u_2}{m_1 + m_2} ),
  ( v_2 = \frac{(m_2 - m_1)u_2 + 2m_1u_1}{m_1 + m_2} ).
• If ( m_1 = m_2 ) and ( u_2 = 0 ), then ( v_1 = 0 ), ( v_2 = u_1 ) (velocity exchange).
• In two-dimensional elastic collision, momentum is conserved along both x and y axes; kinetic energy is also conserved.
• For an isolated system, total momentum is conserved in all collisions; verified from NCERT.
• Mechanical energy is conserved only in the absence of non-conservative forces like friction.
• The area under a force-displacement graph gives work done; e.g., constant force of 10 N over 5 m gives area = 50 J.
• Negative work occurs when force opposes displacement; e.g., kinetic friction always does negative work.

Difficulty Level

Intermediate — requires understanding of vector dot products, conservation laws, and ability to apply formulas in collision problems.

Common CUET Traps

• Trap: Assuming kinetic energy is conserved in all collisions. Avoid: Remember kinetic energy is conserved only in elastic collisions.
• Trap: Using ( W = F s ) without considering angle ( \theta ). Avoid: Always use ( W = F s \cos\theta ), especially when force and displacement are not aligned.
• Trap: Confusing power with energy. Avoid: Power is rate of doing work (( P = W/t )), not total work.

Practice MCQs

1. A force of 20 N acts on a body at an angle of 60° to displacement of 4 m. The work done is:
   A. 40 J
   B. 80 J
   C. 160 J
   D. 20 J
   Answer: A
   Explanation: ( W = Fs\cos\theta = 20 \times 4 \times \cos60^\circ = 80 \times 0.5 = 40 \text{ J} ).
   Why others fail: B ignores the cosine factor and uses ( W = Fs ) directly.

2. A 2 kg block moves with speed 5 m/s. Its kinetic energy is:
   A. 10 J
   B. 25 J
   C. 50 J
   D. 100 J
   Answer: B
   Explanation: ( K = \frac{1}{2}mv^2 = \frac{1}{2} \times 2 \times 25 = 25 \text{ J} ).
   Why others fail: C results from forgetting the ( \frac{1}{2} ) in kinetic energy formula.

3. In a perfectly inelastic collision between two objects of equal mass moving toward each other with equal speed, the final kinetic energy is:
   A. Equal to initial
   B. Half of initial
   C. Zero
   D. One-fourth of initial
   Answer: C
   Explanation: Objects stick together and come to rest due to equal and opposite momenta; total momentum is zero, so final velocity is zero.
   Why others fail: B is chosen by those who confuse with energy loss in general inelastic collisions.

4. A ball falls from height ( h ) and rebounds to height ( 0.64h ). The coefficient of restitution is:
   A. 0.64
   B. 0.8
   C. 0.9
   D. 0.7
   Answer: B
   Explanation: ( e = \sqrt{\frac{h'}{h}} = \sqrt{0.64} = 0.8 ).
   Why others fail: A is chosen by those who use height ratio directly instead of square root.

5. A 10 kg mass moving at 6 m/s collides elastically with a stationary 2 kg mass. The final velocity of the 2 kg mass is:
   A. 8 m/s
   B. 10 m/s
   C. 12 m/s
   D. 14 m/s
   Answer: B
   Explanation: Using ( v_2 = \frac{2m_1u_1}{m_1 + m_2} = \frac{2 \times 10 \times 6}{10 + 2} = \frac{120}{12} = 10 \text{ m/s} ).
   Why others fail: A results from incorrect application of momentum conservation without considering elastic collision formula.

Last‑Minute Revision

• ⚠️ Work is scalar, but sign depends on angle between ( \vec{F} ) and ( \vec{s} ).
• ⚠️ ( W = 0 ) if ( \theta = 90^\circ ) — e.g., Earth’s gravity does no work on satellite in circular orbit.
• ⚠️ ( \Delta K = W_{\text{net}} ) — Work-Energy Theorem applies even with variable forces.
• ⚠️ Conservative force: work path-independent; e.g., gravity.
• ⚠️ Non-conservative force: work path-dependent; e.g., friction.
• ⚠️ ( U_{\text{grav}} = mgh ) only near Earth’s surface.
• ⚠️ ( U_{\text{spring}} = \frac{1}{2}kx^2 ) — x is displacement from equilibrium.
• ⚠️ Power ( P = Fv\cos\theta ) — instantaneous power.
• ⚠️ 1 hp = 746 W — standard conversion.
• ⚠️ In elastic collision: ( \Delta K = 0 ), ( \Delta p = 0 ).
• ⚠️ In inelastic collision: ( \Delta p = 0 ), ( \Delta K \neq 0 ).
• ⚠️ Perfectly inelastic: ( e = 0 ), maximum KE loss.
• ⚠️ ( e = \sqrt{\frac{h_{\text{rebound}}}{h_{\text{fall}}}} ) — for vertical bounce.
• ⚠️ Head-on elastic collision: velocities exchange if masses equal and one initially at rest.
• ⚠️ Momentum conserved in all isolated systems — key for collision problems.
• ⚠️ 2D collision: conserve momentum separately in x and y.
• ⚠️ Area under F-s graph = work done — true for all force types.
• ⚠️ Negative work: force component opposite to displacement.
• ⚠️ Spring force is conservative — mechanical energy conserved in spring-mass system.
• ⚠️ Mnemonic: “Elastic – Energy Conserved; Inelastic – Stick Together” – helps recall collision types.