Prerequisites
Before studying this topic, make sure you understand:
- Motion in a Plane - 2D vector motion
- Kinematic Equations - for uniform acceleration
What is Projectile Motion?
A projectile is any object that is thrown into the air and moves under the influence of gravity alone (no air resistance).
Key Characteristics:
- Horizontal motion: uniform velocity (no acceleration)
- Vertical motion: uniformly accelerated ($a = g$ downward)
- Path: Parabola
The horizontal and vertical motions are completely independent. What happens in x doesn’t affect y, and vice versa.
Movie Connection: In Pathaan (2023), when Shah Rukh Khan jumps from helicopters or between buildings, his horizontal speed stays constant while gravity pulls him down — classic projectile motion!
Interactive Demo
Explore projectile motion with this full-featured simulator. Adjust parameters and observe how they affect the trajectory:
Controls:
- Initial Velocity (u): Drag the slider to set launch speed (10-100 m/s)
- Launch Angle: Set the angle of projection (0-90 degrees)
- Gravity Toggle: Switch between Earth (g = 9.8 m/s squared) and Moon (g = 1.6 m/s squared)
- Play/Pause: Start or pause the animation
- Reset: Return projectile to starting position
What to Observe:
Blue arrow (vx): Horizontal velocity stays CONSTANT throughout the flight
Green arrow (vy): Vertical velocity decreases going up, becomes zero at apex, then increases downward
Red dashed arrow (v): Resultant velocity (vector sum of vx and vy)
Green dot at apex: Maximum height where vy = 0
Orange range marker: Horizontal distance covered
Try These Experiments:
- Set angle to 45 degrees and note the range. Then try 30 degrees and 60 degrees - they have the SAME range!
- Enable “Show complementary angle” to see both trajectories simultaneously
- Switch to Moon gravity - notice how much farther the projectile travels
- At any angle, observe that vx never changes while vy constantly decreases
Types of Projectile Motion
1. Horizontal Projection
Object is thrown horizontally from a height $h$ with velocity $u$.
Initial Conditions:
- $u_x = u$ (horizontal)
- $u_y = 0$ (no vertical component)
Equations:
| Direction | Motion Type | Equations |
|---|---|---|
| Horizontal (x) | Uniform | $x = ut$ |
| Vertical (y) | Accelerated | $y = \frac{1}{2}gt^2$, $v_y = gt$ |
Key Formulas:
$$\boxed{T = \sqrt{\frac{2h}{g}}}$$ $$\boxed{R = u\sqrt{\frac{2h}{g}} = uT}$$ $$\boxed{v_{final} = \sqrt{u^2 + 2gh}}$$2. Oblique Projection (Ground to Ground)
Object is thrown at angle $\theta$ with horizontal, with initial velocity $u$.
Initial Components:
- $u_x = u\cos\theta$
- $u_y = u\sin\theta$
At any time t:
- $x = (u\cos\theta)t$
- $y = (u\sin\theta)t - \frac{1}{2}gt^2$
- $v_x = u\cos\theta$ (constant)
- $v_y = u\sin\theta - gt$
Key Formulas for Oblique Projection
Time of Flight
$$\boxed{T = \frac{2u\sin\theta}{g}}$$Time to reach maximum height = $\frac{T}{2} = \frac{u\sin\theta}{g}$
Maximum Height
$$\boxed{H_{max} = \frac{u^2\sin^2\theta}{2g}}$$Horizontal Range
$$\boxed{R = \frac{u^2\sin 2\theta}{g}}$$Equation of Trajectory
$$\boxed{y = x\tan\theta - \frac{gx^2}{2u^2\cos^2\theta}}$$This is a parabola (quadratic in x).
Alternative form: $y = x\tan\theta\left(1 - \frac{x}{R}\right)$
Important Properties
Maximum Range
Range is maximum when $\sin 2\theta = 1$, i.e., $\theta = 45°$
$$R_{max} = \frac{u^2}{g}$$Complementary Angles
For angles $\theta$ and $(90° - \theta)$:
- Same range (since $\sin 2\theta = \sin 2(90° - \theta)$)
- Different heights and times
| Angle | Range | Height | Time |
|---|---|---|---|
| $30°$ | $R$ | $H_1$ | $T_1$ |
| $60°$ | $R$ (same) | $H_2 > H_1$ | $T_2 > T_1$ |
Velocity at Any Point
At height $h$:
$$v = \sqrt{u^2 - 2gh}$$At any time:
$$v = \sqrt{v_x^2 + v_y^2} = \sqrt{u^2\cos^2\theta + (u\sin\theta - gt)^2}$$Velocity Direction
$$\tan\alpha = \frac{v_y}{v_x} = \frac{u\sin\theta - gt}{u\cos\theta}$$where $\alpha$ is angle with horizontal.
Don’t confuse:
- Height at time $t$: $y = u_y t - \frac{1}{2}gt^2$
- Maximum height: $H_{max} = \frac{u^2\sin^2\theta}{2g}$
Special Cases
Projection from a Height
If projectile is launched from height $h$ above ground:
Time of flight: Solve $h + (u\sin\theta)T - \frac{1}{2}gT^2 = 0$
$$T = \frac{u\sin\theta + \sqrt{u^2\sin^2\theta + 2gh}}{g}$$Projection on Inclined Plane
For a plane inclined at angle $\alpha$:
- Range along plane: $R = \frac{2u^2\sin(\theta - \alpha)\cos\theta}{g\cos^2\alpha}$
- Maximum range: at $\theta = \frac{\pi}{4} + \frac{\alpha}{2}$
Summary Table
| Quantity | Horizontal Projection | Oblique Projection |
|---|---|---|
| Initial $v_x$ | $u$ | $u\cos\theta$ |
| Initial $v_y$ | $0$ | $u\sin\theta$ |
| Time of flight | $\sqrt{\frac{2h}{g}}$ | $\frac{2u\sin\theta}{g}$ |
| Max height | — | $\frac{u^2\sin^2\theta}{2g}$ |
| Range | $u\sqrt{\frac{2h}{g}}$ | $\frac{u^2\sin 2\theta}{g}$ |
Practice Problems
A ball is thrown horizontally from a 45 m high cliff with speed 20 m/s. Find time of flight and range. (g = 10 m/s²)
Solution: $T = \sqrt{\frac{2h}{g}} = \sqrt{\frac{2 \times 45}{10}} = \sqrt{9} = $ 3 s
$R = uT = 20 \times 3 = $ 60 m
A projectile is launched at 30° with 40 m/s. Find range, max height, and time of flight. (g = 10 m/s²)
Solution: $T = \frac{2u\sin\theta}{g} = \frac{2 \times 40 \times 0.5}{10} = $ 4 s
$H_{max} = \frac{u^2\sin^2\theta}{2g} = \frac{1600 \times 0.25}{20} = $ 20 m
$R = \frac{u^2\sin 2\theta}{g} = \frac{1600 \times \sin 60°}{10} = \frac{1600 \times 0.866}{10} ≈ $ 138.6 m
For what angle of projection is range equal to maximum height?
Solution: $R = H_{max}$ $\frac{u^2\sin 2\theta}{g} = \frac{u^2\sin^2\theta}{2g}$ $2\sin\theta\cos\theta = \frac{\sin^2\theta}{2}$ $4\cos\theta = \sin\theta$ $\tan\theta = 4$ $\theta = \tan^{-1}(4) ≈ $ 76°
Related Topics
Within Kinematics
- Kinematic Equations — The foundation for projectile formulas
- Motion in a Plane — 2D vectors needed here
- Relative Velocity — Projectiles observed from moving frames
- Circular Motion — Another type of 2D motion
Connected Chapters
- Newton’s Laws — Forces during projectile motion
- Work-Energy — Energy approach to projectile problems
- Gravitation — Why projectiles follow parabolic paths
Math Connections
- Trigonometry — Sin, cos for component resolution
- Parabola — The shape of projectile trajectory
- Differentiation — For max height, range optimization