Class 11 · CBSE / NCERT · Physics

Class 11 Physics — Derivations

The important Class 11 derivations, one step at a time. Predict the next line yourself, then reveal step by step — active recall beats re-reading every time.

💡 How to use: read the aim, predict the next step in your head, then click “Reveal step”. Repeat. Use “Show whole derivation” for revision.

Motion in a Straight Line

First equation of motion (v = u + at)

Aim: Derive v = u + at by calculus.

Acceleration a = dv/dt (constant). Definition of acceleration.
dv = a·dt. Rearrange.
Integrate from u to v and 0 to t: ∫dv = a∫dt ⇒ v − u = at. a is constant.

Second equation of motion (s = ut + ½at²)

Aim: Derive s = ut + ½at².

Velocity v = ds/dt = u + at. Using the first equation.
ds = (u + at)·dt. Rearrange.
Integrate from 0 to s and 0 to t: s = ∫(u + at)dt = ut + ½at². Integrate each term.

Motion in a Plane

Range of a projectile

Aim: Show that the horizontal range R = u²sin2θ / g.

Horizontal velocity u_x = u cosθ (constant); vertical u_y = u sinθ. Resolve the launch velocity.
Time of flight T = 2u sinθ / g. Time to go up and come back to launch height.
Range R = u_x · T = (u cosθ)(2u sinθ / g). Horizontal distance in time T.
R = u²·(2 sinθ cosθ) / g = u² sin2θ / g. Use 2 sinθ cosθ = sin2θ.

Work, Energy and Power

Work–energy theorem

Aim: Show that the work done equals the change in kinetic energy.

W = ∫F dx = ∫ma dx. Newton's second law.
a dx = (dv/dt) dx = v dv. Chain rule: dx/dt = v.
W = ∫ m v dv, from u to v. Change the variable of integration.
W = ½mv² − ½mu². Evaluate the integral.

Gravitation

Escape velocity

Aim: Show that the escape velocity v_e = √(2gR).

To escape, the kinetic energy must equal the work done against gravity to infinity. Total energy becomes zero at infinity.
½mv_e² = GMm / R. PE at the surface (magnitude).
v_e² = 2GM / R. Cancel m and rearrange.
Since g = GM/R², GM = gR² ⇒ v_e² = 2gR. Write in terms of g.

Orbital velocity of a satellite

Aim: Show that v_o = √(GM/r).

Gravitational pull provides the centripetal force: GMm/r² = mv_o²/r. Circular orbit of radius r.
Cancel m and one r: GM/r = v_o². Simplify.

Oscillations

Time period of a simple pendulum

Aim: Show that T = 2π√(L/g).

Restoring force for a small displacement: F = −mg sinθ ≈ −mgθ. For small angles sinθ ≈ θ.
With θ = x/L: F = −(mg/L)·x, so a = −(g/L)·x. This is SHM: a = −ω²x.
Compare: ω² = g/L ⇒ ω = √(g/L). Angular frequency of the SHM.
T = 2π/ω. Period–frequency relation.