Projectile Motion: The Curve That Rules Your World

What is it? Anytime you throw, kick, spray, or drop something and let gravity do the rest, that thing is a projectile. It does NOT fly straight. It always makes a smooth curve called a parabola.

Think of:

• A cricket six by Dhoni – ball up, then down
• Water from a garden hose – same curve
• Your phone slipping from your hand – half a curve
• A basketball free throw – perfect parabola

Big idea in one line: Every throw is two separate stories happening at the same time – one sideways, one up-down.

The Secret – Two Motions in One

Imagine you throw a ball at an angle. Your hand gives it one push. We split that push:

1. Sideways motion (X)

No force after you let go (ignore air). So speed stays constant. Like a car on cruise control.

ux = u × cosθ

2. Up-down motion (Y)

Gravity pulls down at 9.8 m/s² every second. So the ball slows going up, stops for a blink, then speeds down.

uy = u × sinθ

Layman trick: Close one eye. Watch only the shadow on the ground – it moves at steady speed. Now watch only the height – it goes up then down like a lift with broken brakes.

Because they are independent, we can solve them separately, then combine.

All Derivations – No Magic, Just Logic

Starting tools (from Class IX)

v = u + at
s = ut + ½ at²
v² = u² + 2as

Step 1 – Position at any time t

Horizontal:
x = ux × t = (u cosθ)t    [no acceleration]

Vertical:
y = uy × t – ½gt² = (u sinθ)t – 4.9t²    [a = -g]

Step 2 – Time of Flight

When it lands, y = 0

0 = uy t – ½gt²

t(uy – ½gt) = 0

T = 2uy/g = 2u sinθ/g

Step 3 – Maximum Height

At top, vertical speed = 0

0 = uy² – 2gH

H = uy²/2g = u²sin²θ/2g

Step 4 – Horizontal Range

Range = horizontal speed × total time

R = ux × T

R = (u cosθ)(2u sinθ/g)

R = u²sin2θ/g

Quick check: Throw at 20 m/s, 45°. Range ≈ 40.8 m. At 30°, range drops. At 60°, same range as 30°. Best angle for distance is 45°.

Link With Real Life and Other Subjects

Values you feel

• Sports: Javelin throwers release near 36°. Basketball arcs are usually above 50°.
• Home: A water pipe at 45° reaches the farthest.
• Safety: Never throw keys straight up. Time up = time down.

Cross-subject links

• Maths: The path is a parabola.
• Biology: Kangaroos optimize jump angles.
• History: Galileo explained projectile motion in 1638.
• Art & PE: Draw and physically experience the curve.
• Life lesson: The ball rises before it travels far — growth before success.

Make It Experiential – 3 Activities

Activity 1 – Table Top Shooter

Roll a marble off a table. Measure height and landing distance. Calculate speed using projectile equations.

Activity 2 – Hose Angle Test

Spray water at different angles and compare ranges. 45° reaches farthest.

Activity 3 – Phone Slow-Mo Lab

Use slow-motion video to observe constant horizontal motion and changing vertical motion.

Safety: Use soft balls and open spaces only.

Question Bank – Test Yourself

1. Why do a dropped ball and a horizontally thrown ball hit together?
2. A cricketer hits at 25 m/s and 60°. Find horizontal speed.
3. Why is 45° best for maximum range?
4. You throw a stone upward at 10 m/s. Time to reach top?
5. Draw velocity vectors at launch, top, and landing.
6. Which other angle gives same range as 30°?
7. Explain projectile motion to a child.
8. Why do long jumpers run before jumping?
9. How does lower gravity on Moon affect range?
10. Design a simple experiment to prove horizontal motion is uniform.
11. Name careers using projectile motion daily.
12. Challenge: A ball thrown from 2 m height lands 30 m away at 45°. Estimate initial speed.