Mechanics & motion

Scalars have size only (distance, speed); vectors have size and direction (displacement, velocity). $\text{speed}=\frac{\text{distance}}{\text{time}}$, $\text{acceleration}=\frac{\text{change in velocity}}{\text{time}}$.

On a distance–time graph the gradient is speed. On a velocity–time graph the gradient is acceleration and the area underneath is distance. The equation of motion (constant acceleration): $v^2-u^2=2as$.

Forces include weight, normal contact, drag/air resistance, friction, magnetic, electrostatic, thrust, upthrust, lift and tension. A force diagram shows them as arrows. The resultant is the single force equivalent to them all; in one dimension, add forces in one direction and subtract the other.

• First law: a body stays at rest or moves at constant velocity unless acted on by a resultant force (its reluctance to change is inertia).
• Second law: $F=ma$; a resultant force causes acceleration.
• Third law: every force has an equal and opposite reaction force *of the same type* on the other body.

Mass (kg) is the amount of matter; weight (N) is the gravitational force on it: $w=mg$, with $g\approx 10\text{ N/kg}$ on Earth. A falling object speeds up until air resistance balances its weight; the resultant force is then zero and it falls at constant terminal velocity.

Hooke's law: $F=kx$. Extension is proportional to force up to the limit of proportionality. Stretch a spring too far (past its elastic limit) and it deforms permanently (inelastic). The energy stored is $E=\frac{1}{2}Fx=\frac{1}{2}k{x}^2$.

Momentum $p=mv$ (a vector). It is conserved in collisions: total momentum before = total after. Force is the rate of change of momentum.

• Confusing mass (kg, constant) with weight (N, depends on $g$).
• Reading the area of a distance–time graph as distance (only velocity–time area is distance).
• Forgetting momentum is a vector (opposite directions need opposite signs).