Particle Kinematics
Kinematics describes motion without considering forces. For rectilinear (straight-line) motion with constant acceleration:
- v = v0 + a*t
- s = s0 + v0*t + (1/2)*a*t^2
- v^2 = v0^2 + 2*a*(s - s0)
For curvilinear motion, components can be expressed in rectangular (x, y), normal-tangential (n, t), or polar (r, theta) coordinates.
Projectile Motion
A projectile moves under gravity alone after launch (neglecting air resistance):
- Horizontal: x = v0*cos(theta)*t, with zero horizontal acceleration
- Vertical: y = v0*sin(theta)*t - (1/2)*g*t^2
- Maximum height: h = v0^2*sin^2(theta)/(2g)
- Range on level ground: R = v0^2*sin(2*theta)/g
- Maximum range occurs at theta = 45 degrees
Newton's Laws of Motion
Newton's second law relates net force to acceleration:
- F = m*a (for a particle of constant mass)
- In normal-tangential coordinates: F_t = m*a_t (tangential), F_n = m*v^2/rho (normal)
- Weight: W = m*g (gravitational force)
- Always draw a free body diagram and kinetic diagram when solving
Work and Energy Methods
Energy methods relate forces to displacement without needing to find acceleration:
- Work of a constant force: U = F*d*cos(theta)
- Work-energy theorem: T1 + Sum(U_1to2) = T2
- Kinetic energy: T = (1/2)*m*v^2
- Potential energy of gravity: V_g = m*g*h (measured from a datum)
- Elastic potential energy: V_e = (1/2)*k*x^2
- Conservation of energy (conservative forces only): T1 + V1 = T2 + V2
- Power: P = F*v = dU/dt
Impulse and Momentum
Useful when forces act over short time intervals or when force-time history is known:
- Linear impulse: integral of F dt = m*v2 - m*v1
- Conservation of linear momentum: when external impulse is zero, m1*v1 + m2*v2 = m1*v1' + m2*v2'
- Direct central impact: coefficient of restitution e = (v2' - v1')/(v1 - v2)
- Perfectly elastic (e = 1): kinetic energy is conserved
- Perfectly plastic (e = 0): objects stick together
Vibrations
Free vibration of a single-degree-of-freedom system:
- Natural frequency: omega_n = sqrt(k/m) (rad/s)
- Natural frequency in Hz: f_n = omega_n / (2*pi)
- Period: T = 1/f_n = 2*pi/omega_n
- Equation of motion: m*x'' + c*x' + k*x = 0 (undamped when c = 0)
- Damping ratio: zeta = c / (2*sqrt(k*m))
- Underdamped (zeta < 1): oscillatory; Overdamped (zeta > 1): no oscillation; Critically damped (zeta = 1)