Key Concepts and Formulas

Scalars and Vectors

  • Scalars: Quantities with magnitude only (e.g., distance, speed, mass).
  • Vectors: Quantities with both magnitude and direction (e.g., displacement, velocity, force). They obey the laws of vector algebra.

Vector Operations

  • Magnitude of a Vector: ∣A⃗∣=Ax2+Ay2+Az2|\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}.
  • Resultant of Two Vectors: ∣A⃗+B⃗∣=A2+B2+2ABcos⁡(θ)|\vec{A} + \vec{B}| = \sqrt{A^2 + B^2 + 2AB\cos(\theta)}.
  • Resolution of a Vector: Ax=Acos⁡(θ)A_x = A \cos(\theta), Ay=Asin⁡(θ)A_y = A \sin(\theta).
  • Scalar (Dot) Product: A scalar quantity given by A⃗⋅B⃗=ABcos⁡(θ)=AxBx+AyBy+AzBz\vec{A} \cdot \vec{B} = AB\cos(\theta) = A_x B_x + A_y B_y + A_z B_z. It is used to find the work done or the angle between vectors.
  • Vector (Cross) Product: A vector quantity with magnitude ∣A⃗×B⃗∣=ABsin⁡(θ)|\vec{A} \times \vec{B}| = AB\sin(\theta) and direction perpendicular to the plane of A and B, given by the right-hand rule. It is used to find torque and angular momentum.

Projectile Motion (launched from ground)

  • Equation of Trajectory: y=(tan⁡θ0)x−g2(v0cos⁡θ0)2x2y = (\tan\theta_0)x - \frac{g}{2(v_0\cos\theta_0)^2}x^2
  • Time of Flight (TfT_f): Tf=2v0sin⁡θ0gT_f = \frac{2v_0\sin\theta_0}{g}
  • Maximum Height (HmaxH_{max}): Hmax=(v0sin⁡θ0)22gH_{max} = \frac{(v_0\sin\theta_0)^2}{2g}
  • Horizontal Range (R): R=v02sin⁡(2θ0)gR = \frac{v_0^2\sin(2\theta_0)}{g}

Uniform Circular Motion (UCM)

  • Centripetal Acceleration (aca_c): ac=v2r=ω2ra_c = \frac{v^2}{r} = \omega^2 r
  • Relation between Linear and Angular Velocity: v=ωrv = \omega r
  • Time Period (T): T=2πωT = \frac{2\pi}{\omega}