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B3.3 · Describe horizontal and vertical characteristics of projectile motion

Learn to describe horizontal and vertical characteristics of projectile motion through clear examples and targeted practice.

Ontario Grade 11 Physics

Kinematics

Describing two parts of one motion

A projectile is an object moving through the air after it is launched or released. In this lesson, the projectile is the system, and the reference frame is fixed to the ground. A reference frame is the viewpoint used to describe position and motion. We ignore air resistance, so gravity is the only force considered during flight. Choose right as positive horizontally and up as positive vertically.

A vector has both magnitude and direction. Velocity and acceleration are vectors. A scalar has magnitude only; time is a scalar. The motion can be described using horizontal and vertical components. A component is one part of a vector along a chosen direction. These components describe the same projectile at the same time.

What you will learn

Describe horizontal motion when air resistance is ignored.
Describe how gravity affects vertical motion.
Explain how horizontal and vertical motion occur at the same time.
Use direction and signs consistently when describing motion.

1. Prerequisite bridge: directions, velocity, and acceleration

Position tells where an object is relative to a chosen starting point. Displacement is the change in position, including direction. With right and up chosen as positive, a displacement to the left or downward is negative in the corresponding direction.

Velocity describes how position changes and includes direction. The horizontal velocity component,

v_x

, describes motion left or right. The vertical velocity component,

v_y

, describes motion up or down. A positive or negative sign reports direction relative to the chosen positive direction; it does not by itself say whether the object is speeding up or slowing down.

Acceleration describes a change in velocity. Near Earth's surface, gravity produces downward acceleration. The symbol

g

is the magnitude of this acceleration, about

9.80\ \mathrm{m/s^2}

. Since up is positive here, the vertical acceleration is negative. The units mean metres per second of velocity change each second.

a_x=0,\qquad a_y=-g

Velocity and acceleration are vectors; time is a scalar.
The chosen positive directions determine the signs of components.
Gravity points down, so vertical acceleration is negative when up is positive.

2. The physical situation and component diagram

A projectile follows a curved path because its horizontal and vertical motions combine. In the no-air-resistance model, gravity changes the vertical motion but does not change the horizontal motion. This model simplifies the situation by leaving air resistance out.

A horizontal launch begins with horizontal velocity and no vertical velocity. The object starts moving downward as gravity acts. An angled launch has both horizontal and vertical velocity at the start. The component descriptions remain separate, even though they happen together.

For an angled launch, picture a rightward arrow labelled

v_x

and an upward arrow labelled

v_y

at the same point. Together, these arrows describe the projectile's velocity at launch. Later in the flight, the horizontal component stays unchanged in this model. The vertical component decreases while the projectile rises, is momentarily zero at its highest point, and points downward after it passes that point.

At the highest point, only the vertical component is momentarily zero. If the projectile has horizontal motion, it continues moving horizontally. Gravity also continues to act there, so vertical acceleration is still downward.

Horizontal acceleration is zero in the model, so horizontal velocity is constant.
Vertical acceleration is constant and downward.
At the highest point, vertical velocity is momentarily zero; horizontal velocity may remain.

3. Governing relationships and motion descriptions

The component relationships summarize the model. Here,

x

and

y

describe horizontal and vertical position,

t

is elapsed time, and the subscript

i

means the initial value, at the start of the motion. The symbol

\Delta

means change. Position is measured in metres, time in seconds, velocity in metres per second, and acceleration in metres per second squared.

Horizontally, zero acceleration means the horizontal velocity does not change. As a result, the horizontal displacement grows steadily with elapsed time. Vertically, the constant downward acceleration changes vertical velocity. The vertical position therefore does not change at a steady rate throughout the flight.

For a launch at an angle above horizontal, the initial velocity can be described by horizontal and vertical components. In a right-triangle diagram, the horizontal component lies along the side next to the angle, and the vertical component lies along the side opposite the angle. This describes one velocity vector, not two different launches.

A position-versus-time graph can help describe the difference. Horizontal position changes at a steady rate, so its graph is a straight line. Vertical position changes at a varying rate, so its graph curves. These descriptions are consistent with constant horizontal velocity and changing vertical velocity.

\Delta x=v_{ix}t,\qquad v_y=v_{iy}-gt,\qquad \Delta y=v_{iy}t-\frac{1}{2}gt^2

The same elapsed time applies to both component descriptions.
Use signs that match the chosen directions.
The curved path does not mean that horizontal velocity changes in this model.

4. Keeping both descriptions in view

Horizontal and vertical motion are independent descriptions, not separate events. For example, an object can move right and downward at the same time. In this case its horizontal velocity is positive, while its vertical velocity is negative under the chosen axes.

The word “independent” does not mean the parts happen at different times. The same elapsed time describes both. It means that, in this model, vertical gravity does not alter horizontal velocity, and horizontal motion does not alter the vertical acceleration.

When describing a projectile, state the direction as well as the magnitude of a velocity component. A statement such as “the object is moving at a certain speed” does not say whether it is moving right, left, up, or down. Check each direction against the chosen sign convention.

Both components describe the projectile at the same moment.
Gravity changes vertical velocity, not horizontal velocity, in the model.
A velocity description should include direction.

Worked example

1. Describing a horizontal launch

A ball leaves a ledge horizontally and moves through the air. Describe its horizontal and vertical motion while it is in flight. Ignore air resistance.

Set the system and directions
The system is the ball, viewed from a reference frame fixed to the ground. Right and up are positive. The ball begins with horizontal motion and no vertical velocity.
Describe the horizontal component
With air resistance ignored, there is no horizontal acceleration in this model. The ball's horizontal velocity therefore remains constant and directed to the right. a_x=0, v_x=constant
Describe the vertical component
Gravity acts downward during the entire flight. The ball's vertical velocity starts at zero and becomes increasingly downward as it falls.
$v_{iy}=0,\qquad a_y=-g$

Answer: The ball moves right at constant horizontal velocity while its vertical motion changes downward under gravity.

Check: The horizontal description has no change in velocity. The vertical description begins with zero vertical velocity but does not stay at zero. The directions agree with right and up chosen as positive.

Worked example

2. Describing an angled launch on the way up

A ball is launched up and to the right at an angle. Describe the direction of each velocity component as it rises, and explain what happens to the components at the highest point.

Set the system and directions
The system is the ball, viewed from the ground. Right and up are positive. The launch gives the ball a positive horizontal component and a positive vertical component.
Follow the horizontal component
Horizontal acceleration is zero in the model. The horizontal velocity remains directed to the right and stays constant during the ascent.
$a_x=0,\qquad v_x>0$
Follow the vertical component
Gravity points down while the ball rises. Its upward vertical velocity gets smaller until it is momentarily zero at the highest point. The vertical acceleration remains downward at that point. v_y=0 at the highest point, a_y=-g

Answer: During the ascent, the ball moves right with constant horizontal velocity while its upward vertical velocity decreases. At the highest point, vertical velocity is momentarily zero, but horizontal velocity may remain to the right.

Check: The highest point is not a point where all motion stops. The answer distinguishes vertical velocity from horizontal velocity and keeps gravity directed downward.

Worked example

3. Describing motion after the highest point

A projectile has passed its highest point and is moving to the right as it descends. Describe its horizontal velocity, vertical velocity, and acceleration with right and up positive.

Set the system and directions
The system is the projectile, and the reference frame is fixed to the ground. Right and up are positive. The description applies after the projectile has passed its highest point.
State the horizontal direction
Ignoring air resistance, horizontal velocity remains constant. Since the projectile moves right, its horizontal component is positive.
$v_x>0,\qquad a_x=0$
State the vertical direction
After the highest point, the projectile moves downward. Its vertical velocity is negative because down is opposite to the positive vertical direction. Gravity gives it negative vertical acceleration.
$v_y<0,\qquad a_y=-g$

Answer: The projectile has constant rightward horizontal velocity, downward vertical velocity, zero horizontal acceleration, and downward vertical acceleration.

Check: The signs match the selected axes. Horizontal velocity and vertical velocity have units of metres per second; acceleration has units of metres per second squared. The directions are physically consistent with a rightward, descending path.

Common mistakes and how to avoid them

Assuming that horizontal velocity decreases just because the projectile is falling.

Correction: With air resistance ignored, horizontal acceleration is zero, so horizontal velocity remains constant in this model.

Treating vertical velocity as zero throughout a horizontal launch.

Correction: Only the initial vertical velocity is zero. Gravity changes it during the flight.

Saying the projectile stops at the highest point.

Correction: Only vertical velocity is momentarily zero there. Horizontal velocity may continue.

Thinking a negative vertical sign means an object is slowing down.

Correction: With up positive, a negative vertical velocity means motion downward. A sign gives direction; it does not alone describe a change in speed.

Lesson summary

Describe projectile motion with horizontal and vertical components.
When air resistance is ignored, horizontal acceleration is zero and horizontal velocity is constant.
Gravity produces constant downward vertical acceleration, so vertical velocity changes.
Both components describe the same projectile at the same time.
At the highest point, vertical velocity is momentarily zero, but horizontal velocity may remain.

Check your understanding

Question 1

With right and up positive, a projectile moves to the right in the no-air-resistance model. What happens to its horizontal velocity?

It remains positive and constant.
It becomes zero at the highest point.
It becomes negative because gravity points down.
It increases because the projectile rises.

Show answer and explanation

It remains positive and constant.

Horizontal acceleration is zero in this model, so rightward horizontal velocity remains constant. Gravity changes the vertical component.

Question 2

Which statement describes an angled projectile at its highest point?

Both velocity components are zero.
Vertical velocity is momentarily zero, while horizontal velocity may remain.
Horizontal velocity is zero, while vertical velocity points up.
Vertical acceleration becomes zero.

Show answer and explanation

Vertical velocity is momentarily zero, while horizontal velocity may remain.

Only the vertical velocity is momentarily zero at the highest point. Gravity still gives downward vertical acceleration.

Question 3

With up positive, which direction does a negative vertical velocity indicate?

Upward motion.
Downward motion.
No vertical motion and no acceleration.
Horizontal motion to the left.

Show answer and explanation

Downward motion.

Down is opposite to the chosen positive vertical direction, so downward vertical velocity has a negative sign.

Key terms

Projectile: An object moving through the air after it is launched or released.
Reference frame: The viewpoint used to describe an object's position and motion.
Component: One part of a vector along a chosen direction, such as horizontal or vertical.
Acceleration: A change in velocity, including a change in its magnitude, direction, or both.
Displacement: A change in position that includes direction.
Initial: The value at the start of the motion being described.

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About this lesson and its review

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Physics (SPH3U), expectation B3.3. It is a study resource, not an official curriculum publication.

Before publication, the draft is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability, and then requires administrator approval. Errors can still occur, so corrections are welcomed.

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