Forces, Newton's Laws and Momentum
Resultant Forces and Newton’s First Law
A resultant force is the overall force on an object once all individual forces are combined. Forces are vectors, so direction matters: two forces of 10 N in opposite directions give a resultant of zero; two forces of 10 N in the same direction give a resultant of 20 N.
Newton’s first law (Sir Isaac Newton, 1643–1727, published in Principia Mathematica, 1687) states that an object will remain at rest or continue to move at a constant velocity unless a non-zero resultant force acts on it. In other words: no resultant force means no change in motion. This is also why inertia — an object’s resistance to change in motion — increases with mass.
Newton’s first law: a resultant force of zero means no change in motion. An object at rest stays at rest; a moving object keeps moving at constant velocity. Inertia resists changes — heavier objects have more inertia and are harder to accelerate or stop.
Newton’s Second Law and F = ma
Newton’s second law states that the acceleration of an object is directly proportional to the resultant force on it and inversely proportional to its mass:
$$F = m a$$
Where \(F\) is resultant force in newtons (N), \(m\) is mass in kilograms (kg), and \(a\) is acceleration in m/s2. Rearranging: \(a = F/m\) and \(m = F/a\).
In this core practical, a trolley on a smooth runway is accelerated by hanging masses on a string over a pulley. Light gates measure the time for the trolley to pass through, allowing calculation of velocity and acceleration. The independent variable is the force applied (changed by moving masses from the trolley to the hanger, keeping total mass constant); the dependent variable is acceleration; controlled variables include total system mass, surface friction, and runway angle.
Plotting force (y-axis) against acceleration (x-axis) gives a straight line through the origin, confirming \(F \propto a\). The gradient equals the total mass being accelerated. Keeping total mass constant is essential — if you add mass to the hanger without removing it from the trolley, both force and mass change, so results are meaningless.
A 1500 kg car has a driving force of 6000 N. Friction and air resistance total 1500 N. Calculate the acceleration.
Step 1: Resultant force = 6000 − 1500 = 4500 N
Step 2: \(a = F / m = 4500 / 1500 = 3\) m/s2
Newton’s Third Law
Newton’s third law states that for every force, there is an equal and opposite reaction force. The key points: the two forces are equal in magnitude, opposite in direction, act on different objects, and are the same type of force. A swimmer pushes water backwards (action); the water pushes the swimmer forwards (reaction).
Students often confuse Newton’s third law with balanced forces. Balanced forces act on the same object (e.g. weight and normal reaction on a resting book). Newton’s third law pairs always act on different objects: the book pushes down on the table (gravity transmitted), the table pushes up on the book.
Weight and Terminal Velocity
The weight of an object is the gravitational force on it:
$$W = m g$$
Where \(W\) is weight in newtons, \(m\) is mass in kg, and \(g\) is gravitational field strength (10 N/kg near Earth’s surface). Mass is the amount of matter in an object (kg); weight changes with gravitational field strength; mass does not.
When an object falls through a fluid (air or liquid), drag increases as speed increases. Eventually drag equals weight and the resultant force is zero — the object reaches terminal velocity. A skydiver opening a parachute dramatically increases drag, decelerating to a lower terminal velocity (suitable for landing).
At terminal velocity: weight = drag, resultant force = 0, velocity is constant. Before terminal velocity: weight > drag, resultant force downward, object accelerates. When parachute opens: drag > weight, resultant force upward, object decelerates to a new lower terminal velocity.
Stopping Distance
The total stopping distance of a vehicle has two components. Thinking distance is the distance covered while the driver reacts; it increases with speed and is worsened by tiredness, alcohol, drugs, or distraction. Braking distance is the distance covered while the brakes slow the vehicle; it increases with speed, wet/icy roads, worn tyres, and worn brakes.
Total stopping distance = thinking distance + braking distance. At higher speeds, braking distance increases much more than linearly (because kinetic energy is proportional to \(v^2\)), so doubling speed more than doubles braking distance.
Momentum and Its Conservation
Momentum (\(p\)) is calculated as:
$$p = m v$$
Where \(p\) is momentum in kg m/s, \(m\) is mass in kg, and \(v\) is velocity in m/s. Momentum is a vector — direction matters.
The law of conservation of momentum states that in a closed system (no external forces), total momentum before a collision equals total momentum after. This principle underpins how engineers design car crumple zones, safety barriers, and space probes.
A 1000 kg car moving at 12 m/s collides with and joins to a stationary 500 kg car. Calculate their combined velocity after the collision.
Momentum before = \(1000 \times 12 + 500 \times 0 = 12\,000\) kg m/s
Momentum after = \((1000 + 500) \times v = 1500v\)
Conservation: \(1500v = 12\,000\)
\(v = 12\,000 / 1500 = 8\) m/s