Mathematics · Year 12 · Kinematics and Forces · Summer Term

Vertical Motion Under Gravity

Applying SUVAT equations to objects moving under constant gravitational acceleration.

• National Curriculum Attainment Targets
• A-Level: Mathematics - Kinematics

About This Topic

Forces and Newton's Laws introduces the 'why' behind motion. Students move from describing how things move (kinematics) to explaining what causes that motion (dynamics). This topic covers Newton's three laws of motion, the concept of resultant force, and the use of vector diagrams to resolve forces. It is a fundamental part of the A-Level Mechanics specification.

Students learn to draw Free Body Diagrams to isolate the forces acting on a single object. This skill is essential for solving problems involving friction, tension in strings, and normal reaction forces. The topic also introduces connected particles, such as pulleys or cars towing trailers, requiring students to solve simultaneous equations based on F = ma.

Students grasp this concept faster through collaborative investigations where they can physically model the balance of forces.

Key Questions

1. Analyze the symmetry of vertical motion under gravity.
2. Construct solutions for projectile motion problems neglecting air resistance.
3. Predict the maximum height and time of flight for an object thrown vertically.

Learning Objectives

• Calculate the maximum height reached by an object thrown vertically upwards.
• Determine the total time of flight for an object launched and landing at the same vertical level.
• Analyze the symmetry of an object's vertical motion under gravity, comparing its upward and downward journeys.
• Construct solutions to problems involving objects projected vertically under constant gravitational acceleration, neglecting air resistance.

Before You Start

Introduction to Kinematics (SUVAT)

Why: Students must be familiar with the basic SUVAT equations and their application to motion in one dimension before applying them to vertical motion under gravity.

Vectors and Scalars

Why: Understanding the difference between vector and scalar quantities, particularly displacement and velocity, is crucial for correctly applying the sign conventions in vertical motion problems.

Key Vocabulary

SUVAT equations	A set of five kinematic equations that describe motion with constant acceleration, relating displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t).
Gravitational acceleration (g)	The constant acceleration experienced by an object due to Earth's gravity, approximately 9.8 m/s², directed downwards.
Maximum height	The highest vertical position an object reaches during its trajectory, occurring when its vertical velocity momentarily becomes zero.
Time of flight	The total duration an object spends in the air, from the moment it is projected until it returns to its starting vertical level.
Displacement	The change in position of an object, a vector quantity representing the shortest distance from the initial to the final position.

Watch Out for These Misconceptions

Common Misconception

Thinking a force is needed to keep an object moving at a constant speed.

What to Teach Instead

This is an Aristotelian view. Newton's First Law states that an object will keep moving unless a *resultant* force acts on it. A 'tug-of-war' investigation helps students see that balanced forces result in constant velocity (or rest).

Common Misconception

Assuming the normal reaction force is always equal to 'mg'.

What to Teach Instead

Students often forget that on a slope or when an object is being pulled at an angle, the reaction force changes. Using 'station rotations' with blocks on inclined planes helps them see how the geometry affects the support force.

Active Learning Ideas

Inquiry Circle: The Tug-of-War Vector

Using force meters, students pull on a central ring from three different directions. They must record the forces and use vector addition (drawing or trigonometry) to show that the resultant force is zero when the ring is stationary.

45 min·Small Groups

Station Rotations: Free Body Diagram Clinic

Set up stations with physical setups (e.g., a block on a ramp, a weight hanging from a pulley). Students must draw the Free Body Diagram for each, identifying all forces and their directions before moving to the next station.

40 min·Small Groups

Think-Pair-Share: Newton's Third Law Paradox

Present the classic 'Horse and Cart' problem: if the horse pulls the cart and the cart pulls back with an equal force, how does anything move? Students discuss in pairs and then explain the answer to the class.

20 min·Pairs

Real-World Connections

• Engineers designing amusement park rides, such as drop towers, use these principles to calculate safe descent speeds and forces experienced by riders.
• Athletes in sports like high jump or shot put implicitly utilize these concepts to optimize their technique for maximum height or distance, understanding the trajectory under gravity.
• Ballistics experts analyzing the trajectory of projectiles, like artillery shells or even thrown objects in forensic investigations, apply similar kinematic equations to reconstruct events.

Assessment Ideas

Quick Check

Present students with a scenario: 'A ball is thrown vertically upwards with an initial velocity of 15 m/s. Calculate its velocity after 1 second and its maximum height.' Observe their application of SUVAT equations and correct sign conventions for acceleration.

Exit Ticket

Ask students to write on a slip of paper: 'Describe in your own words why the time taken for an object to go up to its maximum height is equal to the time taken to fall back down to its starting point, assuming no air resistance.' Evaluate their understanding of symmetry in vertical motion.

Discussion Prompt

Pose the question: 'How would the maximum height and time of flight change if we considered air resistance? Which factors would become more significant?' Facilitate a class discussion on the limitations of the current model and introduce the complexity of real-world scenarios.

Frequently Asked Questions

What is a 'resultant' force?

The resultant force is the single force that represents the vector sum of all individual forces acting on an object. If the resultant force is zero, the object is in equilibrium and its velocity will not change.

How does Newton's Third Law apply to a book on a table?

The book pushes down on the table, and the table pushes up on the book with an equal and opposite force. Note that these two forces act on *different* objects, which is why they don't 'cancel out' to prevent the book from existing.

What is the difference between mass and weight?

Mass is the amount of matter in an object (measured in kg) and is constant everywhere. Weight is the force of gravity acting on that mass (measured in Newtons) and changes depending on the local gravitational field strength (W = mg).

How can active learning help students understand Newton's Laws?

Forces are invisible, which makes them hard to learn from a book. Active learning strategies like using force meters or building simple pulley systems allow students to 'feel' the tension and see the acceleration. When students have to resolve vectors physically to predict if a system will move, they build a concrete understanding of F = ma that survives the complexity of exam questions.