Velocity vs Time
v = -Aω sin(ωt) — 90° lag from x.
Key Notes
Velocity in SHM: v(t) = dx/dt = −Aω·sin(ωt + φ).
Maximum velocity v_max = Aω, occurs at equilibrium (x = 0).
Minimum velocity (= 0) at extremes (x = ±A).
Energy view: KE = ½mv² maximum at center, zero at extremes.
v vs x relation: v² = ω²(A² − x²) ⇒ v = ω·√(A² − x²).
v leads x by π/2 in phase (v = Aω cos shifted by 90° from x = A sin).
On v-x phase plot: ellipse with semi-axes A (x) and Aω (v).
Used in calculating KE distribution and timing of SHM events.
Formulas
Velocity (time form)
Derivative of x(t).
Maximum velocity
At equilibrium.
v in terms of x (energy form)
Independent of t; useful for many problems.
Phase relation
v reaches max while x crosses zero.
Important Points
Velocity is MAX at equilibrium (where x = 0) and ZERO at extremes (x = ±A).
v² + ω²x² = ω²A² (constant) — phase-plot ellipse.
If you know v at any point, you can find x: v(x) = ω√(A²−x²).
v_max determines the maximum KE: K_max = ½m·v_max² = ½m·A²ω².
v changes most rapidly at the extremes (where a is maximum).
Common pitfall: thinking v_max occurs at extremes. NO — v_max occurs at equilibrium.
Velocity vs Time notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 11 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.