Cambridge Lower Secondary CheckpointStage 7

Physics: Forces and motion

Science Stage 7 Chapter Notes

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Physics: Forces and motion
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In this chapter

  1. 1Scalars and Vectors: The Basics
  2. 2Describing Motion: Speed, Velocity and Acceleration
  3. 3Graphing Motion: Analysing Journeys
  4. 4Newton's First Law: The Law of Inertia
  5. 5Newton's Second Law: Force, Mass and Acceleration
  6. 6Mass and Weight: What's the Difference?
  7. 7Newton's Third Law: Action and Reaction
  8. 8Momentum and Collisions

1. Scalars and Vectors: The Basics

In physics, we deal with quantities. These can be sorted into two groups: scalars and vectors. A scalar is a quantity that has only a magnitude (a size or amount). Examples include distance (5 metres), speed (10 m/s), mass (70 kg), and time (15 s). A vector is a quantity that has both magnitude and a direction. Examples include displacement (5 metres East), velocity (10 m/s North), force (20 Newtons downwards), and acceleration (2 m/s² forwards). Think of it this way: if you need to point to describe it fully, it's probably a vector.

Key term

Vector: A physical quantity that has both magnitude and direction.

Examiner insight

Examiners award marks for correctly identifying quantities as either scalar or vector, and for using vector addition (like Pythagoras) to find resultant vectors.

Common pitfall

Confusing distance and displacement. Distance is the total path length (a scalar), while displacement is the straight-line separation between the start and end points (a vector).

Worked example 14 marks

A person walks 8 metres East, then turns and walks 6 metres North. Calculate:a) the total distance they have travelled, andb) their final displacement from the starting point.

  1. 1

    a) Distance is a scalar, so we just add the magnitudes of the paths taken. Total distance = 8 m + 6 m = 14 m.

  2. 2

    b) Displacement is a vector, representing the straight-line distance and direction from start to finish. The paths form a right-angled triangle.

  3. 3

    Using Pythagoras' theorem: displacement² = 8² + 6² = 64 + 36 = 100.

  4. 4

    So, the magnitude of the displacement = √100 = 10 m.

  5. 5

    To fully describe the vector, we need the direction. The final displacement is 10 m in a North-East direction from the start.

Recap

  • A scalar quantity has magnitude only.
  • A vector quantity has both magnitude and direction.
  • Distance and speed are scalars.
  • Displacement and velocity are vectors.
  • Vectors can be represented by arrows, where the length shows magnitude and the arrow shows direction.

Quick check

  1. Classify the following as either a scalar or a vector: Mass, Force, Energy, Acceleration.2 marks

2. Describing Motion: Speed, Velocity and Acceleration

Speed is how fast an object is moving. It's a scalar quantity calculated by dividing the distance travelled by the time taken. Velocity is speed in a specific direction, making it a vector. An object's velocity only stays constant if its speed and direction both stay the same. Acceleration is the rate at which an object's velocity changes. Because velocity is a vector, you are accelerating if you speed up, slow down, or change direction. Acceleration is also a vector, measured in metres per second squared (m/s²).

speed = distance / time

average velocity = displacement / time

acceleration = (final velocity - initial velocity) / time

a = (v - u) / t

Key term

Acceleration: The rate of change of velocity per unit of time.

Common pitfall

Forgetting that a change in direction at constant speed (like in circular motion) is also a form of acceleration, because velocity is changing.

Fun fact

The fastest accelerating land animal is the cheetah, which can go from 0 to 60 mph (about 97 km/h) in just 3 seconds, an acceleration greater than many sports cars.

Worked example 12 marks

A sprinter runs a 100 m race in 9.8 s. What is their average speed?

  1. 1

    State the formula: average speed = distance / time.

  2. 2

    Substitute the values: speed = 100 m / 9.8 s.

  3. 3

    Calculate the answer: speed ≈ 10.2 m/s.

Worked example 23 marks

A car accelerates from 10 m/s to 25 m/s in 5 seconds. Calculate its acceleration.

  1. 1

    Identify the initial velocity (u = 10 m/s), final velocity (v = 25 m/s), and time (t = 5 s).

  2. 2

    State the formula: a = (v -u) / t.

  3. 3

    Substitute the values: a = (25 - 10) / 5.

  4. 4

    Calculate the change in velocity: a = 15 / 5.

  5. 5

    Calculate the final answer: a = 3 m/s².

Recap

  • Speed is the rate of change of distance (scalar).
  • Velocity is the rate of change of displacement (vector).
  • Acceleration is the rate of change of velocity (vector).
  • An object can accelerate by changing its speed or its direction.
  • The unit for acceleration is m/s².

Quick check

  1. A cyclist slows down from 12 m/s to 4 m/s in 4 seconds. What is their acceleration? (Note: it will be a negative value, also known as deceleration).2 marks

3. Graphing Motion: Analysing Journeys

Graphs are powerful tools for visualising and analysing motion. There are two main types you need to master. A distance-time graph shows how far an object has moved from its starting point over time. The gradient (steepness) of the line gives the speed. A horizontal line means the object is stationary. A straight, sloped line means constant speed. A curved line means the speed is changing (acceleration). A speed-time graph (or velocity-time graph) shows how an object's speed changes over time. The gradient of this graph gives the acceleration. The area under the line represents the distance travelled.

Gradient of a distance-time graph = Speed

Gradient of a speed-time graph = Acceleration

Area under a speed-time graph = Distance travelled

Key term

Gradient: A measure of the steepness of a line on a graph, calculated as the change in the vertical axis divided by the change in the horizontal axis (rise over run).

Examiner insight

Marks are consistently awarded for correctly calculating the gradient and the area under a speed-time graph. Show your working clearly, including how you read values from the axes.

Common pitfall

Confusing the two types of graphs. A very common mistake is trying to find distance by calculating the area under a distance-time graph.

Worked example 15 marks

The graph shows a car's journey.a) Calculate the acceleration during the first 10 seconds.b) Calculate the total distance travelled in 30 seconds.

  1. 1

    a) Acceleration is the gradient of a speed-time graph. For the first 10 s: Gradient = rise / run = (20 m/s - 0 m/s) / (10 s - 0 s).

  2. 2

    Acceleration = 20 / 10 = 2 m/s².

  3. 3

    b) Distance is the area under the graph. We need to split this into two parts: a triangle (0-10s) and a rectangle (10-30 s).

  4. 4

    Area of triangle = 0.5 × base × height = 0.5 × 10 s × 20 m/s = 100 m.

  5. 5

    Area of rectangle = width × height = (30 s - 10s) × 20 m/s = 20 s × 20 m/s = 400 m.

  6. 6

    Total distance = 100 m + 400 m = 500 m.

Recap

  • On a distance-time graph, the gradient is speed.
  • On a speed-time graph, the gradient is acceleration.
  • On a speed-time graph, the area underneath is the distance travelled.
  • A horizontal line on a distance-time graph means stationary.
  • A horizontal line on a speed-time graph means constant speed.

Quick check

  1. What does a horizontal line on a velocity-time graph tell you about an object's motion?1 mark
  2. How would you find the speed of an object from its distance-time graph?1 mark

4. Newton's First Law: The Law of Inertia

Newton's First Law of Motion states that an object will remain at rest or continue to move at a constant velocity unless acted upon by a resultant force. 'Constant velocity' means constant speed in a straight line. This property of objects to resist changes in their motion is called inertia. The more mass an object has, the more inertia it has. The key idea is 'resultant force'. If all the forces acting on an object are balanced (e.g., the forward thrust of a car's engine is equal to the backward drag and friction), the resultant force is zero. If the resultant force is zero, the object does not accelerate; its velocity remains constant.

If Resultant Force = 0, then acceleration = 0 (velocity is constant).

Key term

Inertia: The tendency of an object to resist changes in its state of motion.

Common pitfall

Believing that a moving object must have a force acting on it to keep it moving. A force is only needed to change its motion (i.e., to accelerate it).

Fun fact

When you're in a fast-moving car that stops suddenly, your body continues to move forward due to its inertia. This is why seatbelts are crucial!

Worked example 13 marks

A car is travelling along a straight, level road at a constant speed of 25 m/s. The driving force from the engine is 1200 N. What is the total magnitude of the resistive forces (like air resistance and friction) acting on the car?

  1. 1

    Identify the key information: 'constant speed'.

  2. 2

    According to Newton's First Law, if the speed (and velocity) is constant, the acceleration is zero.

  3. 3

    If acceleration is zero, the resultant force on the car must be zero.

  4. 4

    This means the forward forces must be balanced by the backward forces.

  5. 5

    Therefore, Total Resistive Forces = Driving Force = 1200 N.

Recap

  • An object's velocity only changes if there is a resultant force.
  • Zero resultant force means constant velocity (which can be zero).
  • Inertia is the resistance of an object to a change in its motion.
  • An object's inertia depends on its mass.
  • Balanced forces mean the resultant force is zero.

Quick check

  1. A satellite orbits the Earth in a circular path at a constant speed. Is there a resultant force acting on it? Explain your answer.2 marks

5. Newton's Second Law: Force, Mass and Acceleration

Newton's Second Law provides the link between force, mass, and acceleration. It states that the acceleration of an object is directly proportional to the resultant force acting on it, and inversely proportional to its mass. This is summed up in the most important equation in mechanics: F = ma. Here, 'F' is the resultant force in Newtons (N), 'm' is the mass of the object in kilograms (kg), and 'a' is the acceleration in metres per second squared (m/s²). Remember, the 'F' in this equation is always the *resultant* force, which is the sum of all forces acting on the object, taking their directions into account.

F = m × a (Resultant Force = mass × acceleration)

Key term

Resultant Force: The single force that has the same effect as all the individual forces acting on an object combined.

Examiner insight

Examiners look for the correct application of F=ma. This includes identifying all forces, calculating the resultant force, stating the formula, and then substituting values. Clear steps are vital.

Common pitfall

Using the driving force instead of the resultant force in the F=ma equation. You must always calculate the net force first by subtracting opposing forces.

Worked example 12 marks

A ball of mass 0.5 kg is kicked and accelerates at 40 m/s². What is the resultant force on the ball?

  1. 1

    State the formula: F = m × a.

  2. 2

    Substitute the known values: F = 0.5 kg × 40 m/s².

  3. 3

    Calculate the result: F = 20 N.

Worked example 24 marks

A car of mass 1000 kg has a driving force of 3000 N from its engine. It experiences resistive forces of 500 N. Calculate the car's acceleration.

  1. 1

    First, calculate the resultant force. The forces are in opposite directions. Resultant Force = Driving Force - Resistive Forces.

  2. 2

    Resultant Force = 3000 N - 500 N = 2500 N.

  3. 3

    Now use Newton's Second Law. State the formula: F = m × a.

  4. 4

    Rearrange the formula to find acceleration: a = F / m.

  5. 5

    Substitute the values: a = 2500 N / 1000 kg.

  6. 6

    Calculate the final answer: a = 2.5 m/s².

Recap

  • Newton's Second Law is summarised by the equation F = ma.
  • 'F' in the equation is the resultant force, not just one of the forces.
  • Acceleration is directly proportional to resultant force.
  • Acceleration is inversely proportional to mass.
  • Ensure you use standard units: Newtons (N), kilograms (kg), and metres per second squared (m/s²).

Quick check

  1. A resultant force of 50 N acts on an object of mass 10 kg. What is its acceleration?2 marks

6. Mass and Weight: What's the Difference?

Mass and weight are often confused, but they are very different. Mass is the amount of matter ('stuff') in an object. It is measured in kilograms (kg) and it is a scalar quantity. An object's mass is constant, no matter where it is in the universe. Weight is the force of gravity acting on an object's mass. It is a force, so it's measured in Newtons (N) and it is a vector (it always acts towards the centre of the planet or star). An object's weight can change depending on the strength of the gravitational field it is in. The relationship is given by the formula W = mg, where 'g' is the gravitational field strength, measured in N/kg.

W = m × g (Weight = mass × gravitational field strength)

Key term

Weight: The force acting on an object due to gravity.

Common pitfall

Using the terms 'mass' and 'weight' interchangeably or using the wrong units for them (e.g., saying 'my weight is 70 kg').

Fun fact

On Jupiter, with its immense gravity, you would weigh about 2.5 times what you do on Earth. However, your mass would be exactly the same!

Worked example 14 marks

An astronaut has a mass of 75 kg. On Earth, the gravitational field strength(g) is 9.8 N/kg. On the Moon, g is 1.6 N/kg. Calculate:a) her weight on Earth,b) her mass on the Moon,c) her weight on the Moon.

  1. 1

    a) Weight on Earth: W = m × g_Earth = 75 kg × 9.8 N/kg = 735 N.

  2. 2

    b) Mass on the Moon: Mass is the amount of matter and does not change with location. Her mass on the Moon is still 75 kg.

  3. 3

    c) Weight on the Moon: W = m × g_Moon = 75 kg × 1.6 N/kg = 120 N.

Recap

  • Mass is the amount of matter in an object, measured in kg.
  • Weight is the force of gravity on an object, measured in N.
  • Mass is constant everywhere; weight depends on the local gravitational field strength (g).
  • The formula linking them is Weight = mass × gravitational field strength (W = mg).
  • On Earth, g is approximately 9.8 N/kg (often rounded to 10 N/kg in exams).

Quick check

  1. An apple has a mass of 0.1 kg. What is its weight on Earth? (Use g = 10 N/kg)2 marks

7. Newton's Third Law: Action and Reaction

Newton's Third Law of Motion describes what happens when two objects interact. It states that for every action, there is an equal and opposite reaction. This means that if object A exerts a force on object B, then object B simultaneously exerts a force on object A that is equal in magnitude and opposite in direction. The most important thing to remember is that these two forces act on *different objects*. Because they act on different objects, they do not cancel each other out. This law explains how rockets move, how you can swim, and why a gun recoils when fired.

Force on A by B = - Force on B by A

Key term

Action-Reaction Pair: A pair of forces acting on two interacting objects that are equal in magnitude and opposite in direction.

Examiner insight

Examiners reward students who can correctly identify an action-reaction pair, specifying the type of force, the object it acts on, and its direction for both forces in the pair.

Common pitfall

Thinking that action-reaction forces cancel each other out. They cannot, because they act on different objects and you can only cancel forces acting on the same object.

Worked example 13 marks

A book is resting on a table. Describe the action-reaction force pair involving the book and the table.

  1. 1

    The 'action' force is the weight of the book pushing down on the table. So, Force 1 is: The book exerts a downward force on the table.

  2. 2

    The 'reaction' force is the table pushing up on the book. So, Force 2 is: The table exerts an equal upward force on the book.

  3. 3

    These two forces are equal in magnitude, opposite in direction, and act on different objects (one on the table, one on the book).

Recap

  • Forces always occur in pairs.
  • These pairs are called action-reaction pairs.
  • The two forces in a pair are equal in magnitude.
  • The two forces in a pair are opposite in direction.
  • Crucially, the two forces act on different objects.

Quick check

  1. When a cannon fires a cannonball, the cannon recoils backwards. Explain why, using Newton's Third Law.2 marks

8. Momentum and Collisions

Momentum is a measure of an object's motion, often described as 'mass in motion'. It is a vector quantity calculated by multiplying an object's mass by its velocity. The formula is p = mv. A key principle in physics is the Conservation of Momentum. It states that for a closed system (one with no external forces like friction), the total momentum before an event (like a collision or explosion) is equal to the total momentum after the event. This principle is incredibly useful for analysing what happens when objects interact.

p = m × v (momentum = mass × velocity)

Total momentum before collision = Total momentum after collision

Key term

Conservation of Momentum: The principle that in a closed system, the total momentum before an event is equal to the total momentum after the event.

Examiner insight

In collision problems, marks are awarded for a clear statement of the conservation of momentum principle, followed by a correct calculation that treats momentum as a vector quantity (using positive and negative signs for direction).

Common pitfall

Forgetting that momentum is a vector. When objects are moving in opposite directions, one of the velocities must be treated as negative in calculations.

Worked example 14 marks

A trolley of mass 2 kg moving at 5 m/s collides with a stationary trolley of mass 3 kg. After the collision, the two trolleys stick together. Calculate their velocity after the collision.

  1. 1

    State the principle: Total momentum before = Total momentum after.

  2. 2

    Calculate momentum before: Momentum of first trolley = 2 kg × 5 m/s = 10 kg m/s. Momentum of second trolley = 3 kg × 0 m/s = 0. Total momentum before = 10 + 0 = 10 kg m/s.

  3. 3

    Define momentum after: The trolleys stick together, so their combined mass is 2 kg + 3 kg = 5 kg. Let their common velocity be 'v'. Total momentum after = (5 kg) × v.

  4. 4

    Equate before and after: 10 kg m/s = 5v.

  5. 5

    Solve for v: v = 10 / 5 = 2 m/s. The trolleys move together at 2 m/s.

Recap

  • Momentum is a vector quantity calculated as mass times velocity (p = mv).
  • The unit of momentum is the kilogram metre per second (kg m/s).
  • In any collision or explosion, the total momentum is conserved (stays the same).
  • When calculating momentum, remember to account for direction by using positive and negative velocities.

Quick check

  1. What is the momentum of a 1200 kg car travelling at 20 m/s?2 marks

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Distinguish between a scalar quantity and a vector quantity. Give one example of each.2 marks
  2. A lorry has a mass of 15,000 kg. Calculate its weight on Earth, where the gravitational field strength is 9.8 N/kg.2 marks
  3. The velocity-time graph for a cyclist's journey is shown. Calculate (a) the acceleration from 0 to 20 seconds, and (b) the total distance travelled in 60 seconds.4 marks
  4. A car of mass 1200 kg accelerates from rest to 18 m/s in 12 seconds. Calculate (a) the acceleration, and (b) the resultant force required to produce this acceleration.4 marks
  5. A 4 kg bowling ball moving at 3 m/s collides with a stationary 1 kg bowling pin. After the collision, the ball continues to move forward at 2.5 m/s. Calculate the velocity of the pin immediately after the collision.4 marks
  6. Explain the difference between mass and weight. Your explanation should include their definitions, units, and how they are affected by moving from the Earth to the Moon.3 marks
  7. A skydiver of mass 80 kg jumps from a plane. At one point, she is falling at a constant velocity (terminal velocity). If the force of air resistance is 784 N, explain why her velocity is constant, referencing Newton's laws.3 marks
  8. A 100 kg cannon is at rest on a frictionless surface. It fires a 2 kg cannonball horizontally at a velocity of 200 m/s. Calculate the recoil velocity of the cannon.4 marks
  9. A motorbike of mass 200 kg has a driving force of 1500 N. At a certain speed, the total resistive forces are 700 N. Calculate the acceleration of the motorbike at this speed.3 marks
  10. A 5 kg block and a 3 kg block are in contact on a smooth horizontal surface. A horizontal force of 24 N is applied to the 5 kg block, pushing both blocks along. Calculate (a) the acceleration of the system, and (b) the magnitude of the force exerted by the 5 kg block on the 3 kg block.5 marks

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