Question
How do we systematically solve projectile motion problems by decomposing motion into horizontal and vertical components? What formulas apply to each direction?
(CBSE 11, JEE Main, NEET — projectile motion is among the most frequently tested kinematics topics)
Solution — Step by Step
Projectile motion is just two independent motions happening simultaneously:
- Horizontal (-direction): Uniform velocity (no acceleration, ignoring air resistance)
- Vertical (-direction): Uniformly accelerated motion under gravity ( m/s downward)
These two motions share only one variable: time (). This is the link between them.
If a projectile is launched with velocity at angle to the horizontal:
- Horizontal component: (remains constant throughout)
- Vertical component: (changes due to gravity)
Horizontal equations:
Vertical equations:
Time of flight:
Maximum height:
Range:
Maximum range occurs at :
Complementary angle property: and give the same range but different heights and times of flight. For example, and give equal range.
- Set up coordinates: horizontal, vertical upward
- Decompose initial velocity into and
- Identify what is asked (time, height, range, velocity at a point)
- Use vertical equations to find time (usually from a height condition)
- Use that time in horizontal equations to find range or position
- If asked for velocity: find and separately, then
flowchart TD
A["Projectile problem"] --> B["Decompose u into ux and uy"]
B --> C["Horizontal: x = ux × t (no acceleration)"]
B --> D["Vertical: apply kinematics with a = -g"]
C --> E["Find time from vertical condition"]
D --> E
E --> F["Substitute t into horizontal equation"]
F --> G["Find range/position/velocity"]
H["Special cases"] --> I["Horizontal launch: θ = 0, uy = 0"]
H --> J["From height: y₀ ≠ 0"]
H --> K["On inclined plane: resolve g along/perpendicular to incline"]Why This Works
The decomposition works because gravity acts only vertically — it has no horizontal component. So the horizontal motion is completely unaffected by gravity (constant velocity), while the vertical motion is just free fall with an initial vertical velocity. By treating them separately and connecting through time, we convert a 2D problem into two simpler 1D problems.
This is a direct consequence of Newton’s second law applied component-wise: (so ) and (so ).
Common Mistake
The most frequent error: using (total velocity) instead of (vertical component) in the height formula. The formula uses the vertical component. Students who write forget the factor and overestimate the height. Similarly, time of flight depends on , not . Always decompose first, calculate second.
At maximum height, the vertical velocity is zero but the horizontal velocity is still . So the speed at the highest point is , NOT zero. JEE Main tests this almost every year — “What is the speed at the highest point?” Answer: .