In a transverse wave the particles of the medium vibrate perpendicular to the direction of wave propagation (example: a wave on a stretched string, light waves). In a longitudinal wave the particles vibrate parallel to the direction of propagation, forming compressions and rarefactions (example: sound waves in air).

v = f * lambda

Newton's second law: the rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction of the force. F = dp/dt = d(mv)/dt. For constant mass, F = m (dv/dt) = m a.

F = dp/dt = m a

Systematic errors have a definite cause and always shift readings in one direction (e.g. a zero error in an instrument, or a mis-calibrated scale); they can be minimised by correcting the instrument. Random errors occur due to unpredictable fluctuations and vary in size and sign; they are reduced by taking many readings and averaging.

For a product/power, percentage errors add with the powers as weights: (dP/P) = (dA/A) + 2 (dB/B) = 2% + 2(3%) = 2% + 6% = 8%.

dP/P = dA/A + 2 dB/B

Distance is the total length of the actual path travelled; it is a scalar and is always positive. Displacement is the shortest straight-line vector from the initial to the final position; it is a vector and can be positive, negative or zero. Displacement magnitude is always less than or equal to the distance.

On a v-t graph for constant acceleration, the velocity rises linearly from u to v = u + a t. The displacement equals the area under the graph, which is a rectangle (height u, width t) plus a triangle (base t, height a t): s = (u * t) + (1/2)(t)(a t) = u t + (1/2) a t^2.

s = u t + (1/2) a t^2

Using v^2 = u^2 - 2 a s with v = 0: 0 = (20)^2 - 2(5) s, so 10 s = 400, giving s = 40 m.

v^2 = u^2 - 2 a s

Parallelogram law: if two vectors are represented in magnitude and direction by the two adjacent sides of a parallelogram drawn from a point, their resultant is represented by the diagonal from that point. Magnitude: R = sqrt(P^2 + Q^2 + 2 P Q cos(theta)), and the resultant makes an angle alpha with P where tan(alpha) = (Q sin(theta)) / (P + Q cos(theta)).

R = sqrt(P^2 + Q^2 + 2 P Q cos(theta))

In uniform circular motion, centripetal acceleration is the acceleration directed towards the centre of the circle that continuously changes the direction of velocity. Its magnitude is a_c = v^2 / r = omega^2 r, and it always points radially inward (towards the centre).

a_c = v^2 / r = omega^2 r

R = u^2 sin(2 theta) / g = (30)^2 * sin(60 deg) / 10 = 900 * 0.866 / 10 = 77.9 m (about 78 m).

R = u^2 sin(2 theta) / g

Newton's third law: to every action there is an equal and opposite reaction, and the two forces act on different bodies. Example: when we walk, the foot pushes the ground backward (action) and the ground pushes the foot forward (reaction), which moves us ahead.

F(AB) = - F(BA)

By conservation of linear momentum (initial total momentum = 0): m_bullet * v_bullet = m_gun * v_gun. So 0.020 * 400 = 4 * v_gun, giving v_gun = 8 / 4 = 2 m/s (opposite to the bullet).

m1 v1 = m2 v2

Static friction acts on a body at rest and is self-adjusting up to a maximum value (limiting friction) f_s(max) = mu_s N. Kinetic (sliding) friction acts on a moving body and has a nearly constant value f_k = mu_k N. For the same surfaces, mu_k < mu_s, so it is harder to start motion than to keep it going.

f_s(max) = mu_s N ; f_k = mu_k N

On a smooth incline the acceleration along the plane is a = g sin(theta) = 10 * sin(30 deg) = 10 * 0.5 = 5 m/s^2, directed down the incline.

a = g sin(theta)

Work done by a constant force is the dot product of force and displacement: W = F s cos(theta), where theta is the angle between them. (a) W = 0 when theta = 90 degrees (force perpendicular to displacement, e.g. centripetal force). (b) W is negative when theta is obtuse (90 to 180 degrees), e.g. friction opposing motion.

W = F s cos(theta)

A conservative force does work that depends only on the initial and final positions, not on the path, and the work done in a closed loop is zero (example: gravity, spring force). A non-conservative force does path-dependent work and dissipates energy (example: friction, viscous drag).

Work done W = m g h = 100 * 10 * 10 = 10000 J. Power P = W / t = 10000 / 5 = 2000 W = 2 kW.

P = W / t = m g h / t

In an elastic collision both linear momentum and kinetic energy are conserved (e.g. collisions between hard steel balls, or gas molecules). In an inelastic collision momentum is conserved but kinetic energy is not - some is lost as heat, sound or deformation; in a perfectly inelastic collision the bodies stick together.

Torque is the turning effect of a force about an axis, equal to the product of the force and the perpendicular distance of its line of action from the axis: tau = r F sin(theta) = r x F (cross product). SI unit: newton metre (N m). It is a vector along the axis of rotation.

tau = r F sin(theta)

When the net external torque on a system is zero, its total angular momentum L = I omega remains constant. A spinning skater experiences almost no external torque, so I omega is constant. On pulling the arms in, the moment of inertia I decreases, so the angular speed omega increases to keep L constant - the skater spins faster.

I1 omega1 = I2 omega2