What you'll learn
Motion and Newton's laws is the largest topic in CXC CSEC Physics by question count, and it rewards systematic work more than almost any other. It begins with describing motion — distance, displacement, speed, velocity and acceleration — then moves to representing it on graphs, then to explaining it through Newton's three laws, and finally to momentum and its conservation. The graphs are where most marks are available and most are lost, because a distance-time graph and a velocity-time graph look alike but mean entirely different things. By the end of this guide you should be able to distinguish the motion quantities, use the equations of motion, interpret and draw both graph types, state and apply all three of Newton's laws, explain terminal velocity, and apply the principle of conservation of momentum to collisions.
Key terms and definitions
Distance — the total path length travelled, a scalar
Displacement — the straight-line distance from start to finish in a stated direction, a vector
Speed — distance travelled per unit time, a scalar
Velocity — displacement per unit time in a stated direction, a vector
Acceleration — the rate of change of velocity, a vector measured in metres per second squared
Deceleration — negative acceleration, where an object slows down
Uniform velocity — constant speed in a straight line, so acceleration is zero
Inertia — the tendency of an object to resist a change in its state of motion
Momentum — the product of mass and velocity, a vector measured in kilogram metres per second
Impulse — the product of force and the time for which it acts, equal to the change in momentum
Terminal velocity — the constant velocity reached when the resultant force on a falling object becomes zero
Free fall — motion under gravity alone, with acceleration of about 9.8 metres per second squared
Core concepts
Describing motion
Distance and speed are scalars; displacement and velocity are vectors. A car driving once round a circular track returns to its starting point, so its displacement is zero even though it has covered a substantial distance, and its average velocity is zero even though its average speed is not.
Speed equals distance divided by time. Velocity equals displacement divided by time, with a direction stated.
Acceleration is the change in velocity divided by the time taken. Since velocity is a vector, an object can accelerate by changing its speed, its direction, or both — which is why an object moving in a circle at constant speed is accelerating.
A negative acceleration means the object is slowing down, and the units are metres per second squared in every case.
The equations of motion
Three equations describe uniformly accelerated motion, using v for final velocity, u for initial velocity, a for acceleration, t for time and s for displacement.
The first states that v equals u plus a times t.
The second states that s equals u times t plus one half a times t squared.
The third states that v squared equals u squared plus two times a times s. This one is used when time is not given and not required.
Selecting the right equation is the whole skill. Write down the four quantities you have and the one you want; the correct equation is the one containing exactly those five terms, or the one that omits the quantity you neither have nor want.
For free fall, the acceleration is about 9.8 metres per second squared downwards. Taking downwards as positive simplifies most problems, and an object dropped from rest has an initial velocity of zero.
Distance-time graphs
On a distance-time graph, distance is on the vertical axis and time on the horizontal.
A horizontal line means the object is stationary, since distance is not changing.
A straight sloping line means constant speed, and the gradient gives that speed.
A curve of increasing steepness means the object is accelerating; a curve of decreasing steepness means it is decelerating.
To find the speed at a particular instant on a curve, draw a tangent at that point and calculate its gradient.
Velocity-time graphs
On a velocity-time graph, velocity is on the vertical axis.
A horizontal line means constant velocity, not a stationary object. This is the single most common confusion in the topic, and reading the vertical axis label before interpreting any graph prevents it.
A straight sloping line means uniform acceleration, and the gradient gives the acceleration. A line sloping downwards means deceleration.
Crucially, the area under a velocity-time graph gives the distance travelled. For a line made of straight sections, split the area into rectangles and triangles and add them.
A line that crosses the time axis into negative values indicates motion in the opposite direction.
Newton's first law
An object remains at rest, or continues to move at constant velocity in a straight line, unless acted upon by a resultant force.
This means a moving object needs no force to keep moving; it needs a force only to change its motion. In everyday experience friction and air resistance always act, which is why objects appear to need a push to keep going.
Inertia is the tendency of an object to resist a change in its state of motion, and it depends on mass. A loaded truck is harder to start moving and harder to stop than an empty one.
Practical consequences appear regularly in questions: passengers lurch forward when a vehicle brakes suddenly because their bodies continue moving while the vehicle slows, which is why seatbelts are worn.
Newton's second law
The rate of change of momentum of an object is proportional to the resultant force acting on it, and takes place in the direction of that force.
For constant mass this reduces to the familiar statement that resultant force equals mass multiplied by acceleration, with force in newtons, mass in kilograms and acceleration in metres per second squared.
One newton is the force that gives a mass of one kilogram an acceleration of one metre per second squared.
The word resultant is essential. If several forces act, they must be combined first, and only the resultant produces acceleration. If the resultant is zero, the acceleration is zero and the object continues at constant velocity, which is simply the first law again.
Weight is a particular application: weight equals mass multiplied by the acceleration due to gravity. Mass is a scalar measured in kilograms and does not change with location; weight is a vector force measured in newtons and does change.
Newton's third law
For every action there is an equal and opposite reaction.
More precisely, when two objects interact they exert equal and opposite forces on each other. The two forces are always of the same type, act on different objects, and act along the same line.
Because they act on different objects, they never cancel out — a point candidates frequently get wrong. When a person pushes against a wall, the wall pushes back on the person with equal force, but one force acts on the wall and the other on the person, so they cannot be added to give zero.
Examples include a swimmer pushing water backwards while the water pushes the swimmer forwards, and a rocket expelling gas downwards while the gas pushes the rocket upwards.
Terminal velocity
When an object falls through a fluid it initially accelerates, because its weight exceeds the drag force acting upwards.
As the object speeds up, the drag force increases, so the resultant force decreases and the acceleration falls.
Eventually the drag force equals the weight. The resultant force is then zero, and by Newton's first law the object continues at a constant velocity, its terminal velocity.
On a velocity-time graph this appears as a line that is steep at first, becomes progressively less steep, and finally levels off horizontally.
Momentum and its conservation
Momentum equals mass multiplied by velocity, and being a vector it has direction. In one-dimensional problems, take one direction as positive and the other as negative.
The principle of conservation of momentum states that in a closed system, where no external forces act, the total momentum before a collision equals the total momentum after it.
For two objects colliding, the sum of the two momenta before equals the sum after. If the objects stick together after impact, they move with a common velocity and the combined mass is used.
Impulse is force multiplied by time and equals the change in momentum. This explains why crumple zones, airbags and helmets reduce injury: they increase the time over which the momentum change occurs, so the force experienced is smaller for the same change in momentum.
Worked examples
Example 1: Selecting an equation of motion (4 marks)
A car accelerates uniformly from 8 metres per second to 20 metres per second over a distance of 84 metres. Calculate its acceleration.
The known quantities are initial velocity 8 metres per second, final velocity 20 metres per second and displacement 84 metres. Time is neither given nor required, so use the third equation: v squared equals u squared plus two a s.
Substituting gives 20 squared = 8 squared + 2 × a × 84, so 400 = 64 + 168a.
Therefore 168a = 336, and a = 2.0 metres per second squared.
Example 2: Reading a velocity-time graph (4 marks)
A velocity-time graph shows an object accelerating uniformly from rest to 15 metres per second in 6 seconds, then travelling at constant velocity for 10 seconds. Calculate the acceleration and the total distance travelled.
The acceleration is the gradient of the first section, which is the change in velocity divided by time: 15 ÷ 6 = 2.5 metres per second squared.
The distance is the area under the graph. The first section is a triangle of area one half × 6 × 15 = 45 metres. The second section is a rectangle of area 10 × 15 = 150 metres.
The total distance is 45 + 150 = 195 metres.
Example 3: Conservation of momentum (4 marks)
A trolley of mass 2.0 kilograms moving at 3.0 metres per second collides with a stationary trolley of mass 1.0 kilogram. The two stick together. Calculate their common velocity after the collision.
Momentum before the collision is the sum of the two momenta. The moving trolley has momentum 2.0 × 3.0 = 6.0 kilogram metres per second, and the stationary trolley has zero momentum. The total before is therefore 6.0 kilogram metres per second.
After the collision the combined mass is 2.0 + 1.0 = 3.0 kilograms, moving at a common velocity v, so the momentum after is 3.0v.
By conservation of momentum, 3.0v = 6.0, so v = 2.0 metres per second in the original direction of motion.
Common mistakes and how to avoid them
The most damaging error in this topic is misreading a graph type. On a distance-time graph a horizontal line means stationary; on a velocity-time graph it means constant velocity. Read the vertical axis label first, every time.
Students often use the wrong equation of motion because they have not listed what they know. Writing the five symbols and marking which are known takes ten seconds and selects the equation for you.
In Newton's third law questions, many answers claim the equal and opposite forces cancel. They act on different objects and therefore cannot cancel.
Another frequent slip is confusing mass and weight, or giving weight in kilograms. Weight is a force in newtons.
In momentum problems, candidates often ignore direction. Assign a positive direction at the start and treat opposing motion as negative.
Finally, many candidates calculate distance from a velocity-time graph using the gradient rather than the area. Gradient gives acceleration; area gives distance.
Exam technique for "Motion and Newton's Laws"
Write down the known quantities with their symbols before starting any calculation. This organises the problem and makes the correct equation obvious.
For graph questions, decide first whether the question wants a gradient or an area. Those two operations answer almost every graph question in the topic.
When drawing a tangent on a curve, make it long and mark the two points used. Marks are given for the construction.
State the direction whenever a vector answer is required. A velocity or momentum without direction is incomplete.
For explanation questions on the three laws, name the law you are using and then apply it to the specific situation described, rather than simply reciting the statement.
Quick revision summary
Distance and speed are scalars while displacement and velocity are vectors, and acceleration is the rate of change of velocity, so a change of direction alone counts as acceleration. The three equations of motion are v = u + at, s = ut + half a t squared, and v squared = u squared + 2as, with the third used when time is absent. On a distance-time graph the gradient is speed and a horizontal line means stationary; on a velocity-time graph the gradient is acceleration, a horizontal line means constant velocity, and the area beneath gives distance. Newton's first law states that an object continues at rest or at constant velocity unless a resultant force acts, with inertia depending on mass. The second law gives resultant force equals mass times acceleration, and weight equals mass times gravitational acceleration. The third law pairs equal and opposite forces that act on different objects and so never cancel. Terminal velocity is reached when drag equals weight and the resultant force becomes zero. Momentum is mass times velocity and is conserved in collisions, while impulse is force times time and equals the change in momentum, which is why crumple zones and airbags reduce the force experienced.