Projectile Motion

• To treat projectile motion as independent horizontal (uniform) and vertical (accelerated) components.
• Formulas for time of flight, maximum height, and horizontal range for ground-to-ground projection.
• To analyse motion at angle θ with initial speed u and acceleration g downward.
• To solve JEE-level problems including projection from a height and complementary angles giving same range.

Level 1 — Independence of motion

Verbal: A projectile moves under gravity alone (neglect air resistance). Horizontal velocity stays constant; vertical velocity changes at rate g downward.

Symbolic:

• Horizontal: a_x = 0, v_x = u cos θ, x = (u cos θ) t
• Vertical: a_y = −g, v_y = u sin θ − gt, y = (u sin θ) t − ½gt²

At top of trajectory: v_y = 0, but v_x ≠ 0 — velocity is not zero (common trap).

Level 2 — Standard results (level ground)

Complementary angles: R same for θ and (90° − θ) because sin 2θ = sin(180° − 2θ).

Equation of path: y = x tan θ − (gx²)/(2u² cos²θ) — a parabola.

From height h: time to ground from quadratic in t; do not blindly use T = 2u sin θ/g.

NCERT spotlight — Projection from height and oblique cases

When a stone leaves a cliff, vertical motion starts with initial u sin theta and horizontal with u cos theta, but landing height differs from launch height. Time of flight comes from solving y = 0 or y = -h.

Two angles for same range: theta and 90 minus theta give identical range on level ground because sin 2theta = sin(180 - 2theta).

Envelope of trajectories: Maximum range on level ground at 45 degrees for fixed speed. JEE may ask time to reach a given horizontal distance or clearance over a wall.

A ball is thrown at 20 m/s at 30° above horizontal from ground level. Take g = 10 m/s². Find T, H, and R.

Step 1 — u = 20 m/s, θ = 30°, sin 30° = 0.5, cos 30° = √3/2.
Step 2 — T = 2u sin θ/g = 2×20×0.5/10 = 2 s.
Step 3 — H = u² sin²θ/(2g) = 400×0.25/20 = 5 m.
Step 4 — R = u² sin 2θ/g = 400×sin 60°/10 = 40×(√3/2) ≈ 34.6 m.
Step 5 — At t = 1 s (top): v_y = 0, v_x = 20×√3/2 ≈ 17.3 m/s.

Applications — sports and safety

Cricket ball maximum height and range from launch speed and angle inform field placement. Fire hose stream parabolic path must reach upper floor window — solve for required u and theta. Horizontal range on level ground R = u squared sin 2theta / g peaks at 45 degrees; from elevated launch, optimal angle drops below 45 degrees.

Deep dive — equation of trajectory and clearance problems

Trajectory equation y = x tan theta − (g x²)/(2 u² cos² theta) — parabola opening downward; find x when y = 0 gives range except x=0 launch point. Maximum height on incline launched up slope — rotate axes or decompose g along/perpendicular incline modifies effective g components. Clear wall problem: given wall height h distance d from launch, find minimum u and theta to clear — substitute x=d into trajectory solve inequality y >= h. Horizontal projection from cliff initial vy=0, vx=u — time from vertical y = −(1/2)gt² + h0 = 0 landing. Relative projectile two balls thrown — relative acceleration zero if both only gravity → relative motion straight line JEE trick. Time of flight symmetry level ground: time up equals time down for symmetric parabola; speed magnitude at same height equal going up vs down vertical component opposite horizontal same.

Review and practice drill

Review checklist: (1) Horizontal and vertical motion independent. (2) At apex vertical velocity zero not total velocity. (3) Time of flight from vertical equation. (4) Range formula level ground u squared sin 2 theta over g. Practice: u=20 m/s, theta=30, g=10 — T=2 s, H=5 m, R about 34.6 m.

• For fixed u, which two angles give the same range on level ground?
• A projectile has u = 10 m/s, θ = 45°, g = 10. Find R.
• Why is horizontal acceleration zero for a projectile?

Open the Practice tab for graded questions on Projectile.

• Use the platform-native live simulation or PhET-style tool for this topic (number line, Venn, physics playground, molecule builder, sensor dashboard, etc.).
• Mirror / body / home activity: physically do the concept (count objects, measure, role-play) and photograph or describe for portfolio.
• Voice or text reflection with AI Mentor: explain the concept to a younger student or family member.

• "Explain this concept to a Class 6 student using one real example from an Indian home, school, market, or festival."
• "What is one common mistake students make here, and how would you catch yourself making it?"
• Stretch: "How does this connect to coding, robotics, money, health, environment, or a future career?"

• Complete the core practice + one extension activity (photo, table, short reflection, or mini-project) for base XP + topic badge.
• 5-7 day streak or family discussion note = multiplier + visible artifact in parent/principal dashboard.
• Best real-world application stories (anonymised) featured on class or national leaderboard.

• One hands-on project or measurement using the Drishti kit or household items that makes the concept physical.
• Direct link to at least one Future Skill track (Money Management, Green Tech, Cyber Defenders, Micro-Entrepreneurship, AI Mastery, Sustainable Living, Personality Development).
• Coding extension where relevant (simple script, simulation, or data logging).

This material emphasises experiential "learning by doing", competency (apply/create/analyse), vocational exposure, critical thinking, and multidisciplinary connections. Designed to feed live worlds, AI Mentor (with memory), gamification, robotics, parent analytics, and future skills — not just exam prep.

Portfolio Evidence Idea: Your photo/table/reflection/project + one sentence on "How this helps me in real life or a possible future path."

Open the Practice tab for aligned questions (easy/medium/hard + case-based) with full AI scaffolding.

See curriculum for cross-links and the full future-skills/robotics chapters.