6.2MechanicsStretch

Horizontal and Vertical Components

When a projectile is launched at an angle, its initial velocity splits into horizontal and vertical components. These components then evolve independently — horizontal constant, vertical changing under gravity.

24 min Projectiles

Edexcel A Level Maths: 6.2 Projectiles (Horizontal and Vertical Components)Curated by Kepler, not produced by us. Watch it before the notes below.

Open on YouTube

What you'll be able to do

Resolve the launch velocity into components
Use u cos θ and u sin θ
Track each component over time
Find velocity at a later instant

Initial components

A speed $u$ at angle $\theta$ has horizontal component $u\cos\theta$ (constant) and vertical component $u\sin\theta$ (decreasing under gravity, then increasing downward).

u_x = u\cos\theta, \quad u_y = u\sin\theta

Resolving the launch velocity.

Example 1A ball is launched at 25 m/s at 40°. Find the initial components.

1Horizontal:

25\cos40° \approx 19.2

m/s.

2Vertical:

25\sin40° \approx 16.1

m/s.

Answer

H ≈ 19.2 m/s, V ≈ 16.1 m/s.

Tip — The horizontal component stays the same throughout the flight.

Velocity later on

The horizontal component is unchanged; the vertical component changes by $-gt$ (taking up as positive). Combine them to get the speed and direction at any time.

Formula recap

u_x = u\cos\theta

Horizontal (constant).

v_y = u\sin\theta - gt

Vertical (changes with time).

Common mistakes to avoid

Letting the horizontal component change with time.

Only the vertical component changes (under gravity).

Swapping cos and sin for the components.

Horizontal uses cos θ; vertical uses sin θ.

Key takeaways

Resolve: uₓ = u cos θ, u_y = u sin θ.
Horizontal component is constant.
Vertical component changes by −gt.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

The horizontal component of launch velocity is:
- $u\cos\theta$
- $u\sin\theta$
- $u\tan\theta$
- $u$
Show answer and explanation
Answer: $u\cos\theta$
Adjacent uses cos.
The vertical component is:
- $u\sin\theta$
- $u\cos\theta$
- $u$
- $0$
Show answer and explanation
Answer: $u\sin\theta$
Opposite uses sin.
During flight the horizontal component:
- stays constant
- decreases
- increases
- reverses
Show answer and explanation
Answer: stays constant
No horizontal force.
For 25 m/s at 40°, horizontal component ≈
- $19.2$ m/s
- $16.1$ m/s
- $25$ m/s
- $12.5$ m/s
Show answer and explanation
Answer: $19.2$ m/s
25cos40°.
The vertical velocity at time t (up positive) is:
- $u\sin\theta - gt$
- $u\cos\theta$
- $u\sin\theta + gt$
- $gt$
Show answer and explanation
Answer: $u\sin\theta - gt$
Gravity decelerates upward motion.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

All 4 lessons

Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.