Physics

Class 9

Class 11

Interactive Science Lab

Projectile Motion Simulator

Simulate 2D trajectory motion, launch angles, velocity vectors (Vx, Vy), and range.

Projectile Motion Simulator

Simulate two-dimensional motion under gravity. Explore velocity vectors, time of flight, maximum height, and horizontal range.

Class 9 & 11 Physics

Kinematics

Live Trajectory Canvas(2D Kinematics)

Time (t):0 s/ 4.329 s

Position:x: 0my: 0m

Speed:30 m/s(45°)

0s4.329s

Speed:

Launch Parameters

Quick Preset Scenarios

Initial Speed (v₀)

m/s

0 m/s50 m/s100 m/s

Launch Angle (θ)

deg

0° to 90°

Initial Height (h₀)

0 m (Ground)30 m60 m (Cliff)

Environment / Gravity (g)9.8 m/s²

Karachi Board standard: g = 9.8 m/s² (or 10 m/s² in MCQs).

Launch Parameters

Quick Preset Scenarios

Initial Speed (v₀)

m/s

0 m/s50 m/s100 m/s

Launch Angle (θ)

deg

0° to 90°

Initial Height (h₀)

0 m (Ground)30 m60 m (Cliff)

Environment / Gravity (g)9.8 m/s²

Karachi Board standard: g = 9.8 m/s² (or 10 m/s² in MCQs).

Derived Physics Results

SI Units (m, s, J)

91.837

meters

v₀x × T

22.959

meters

h₀ + v₀y² / 2g

4.329

seconds

Total air time

t_p

2.165

seconds

v₀y / g

v_f

m/s

√(vx² + vy²)

θ_f

45°

below horizontal

tan⁻¹(|vy| / vx)

Horizontal Velocity (v₀x)

21.213 m/s

Constant throughout (no air resistance, a_x = 0).

Initial Vertical Velocity (v₀y)

21.213 m/s

Decreases by 9.8 m/s every second.

Total Energy (E_total)

450 Joules

KE: 450 J + PE: 0 J

Syllabus Step-by-Step Mathematical Derivations

Class 9 & 11 Physics (Karachi / Sindh Textbook Board)

Step 1[m/s]

Resolve Initial Velocity into Components

v_{0x} = v_0 \cos\theta, \quad v_{0y} = v_0 \sin\theta

Sub: v_{0x} = 30 \times \cos(45^\circ) = 21.21, \quad v_{0y} = 30 \times \sin(45^\circ) = 21.21

Calculated:v_{0x} = 21.21 \text{ m/s}, \quad v_{0y} = 21.21 \text{ m/s} m/s

Step 2[s]

Time to Reach Maximum Height (t_p)

t_p = \frac{v_{0y}}{g}

Sub: t_p = \frac{21.21}{9.8}

Calculated:2.16 s

Step 3[m]

Maximum Height from Ground (H_{max})

H_{max} = h_0 + \frac{v_{0y}^2}{2g}

Sub: H_{max} = 0 + \frac{(21.21)^2}{2 \times 9.8} = 0 + 22.96

Calculated:22.96 m

Step 4[s]

Total Time of Flight (T)

T = \frac{2 v_0 \sin\theta}{g}

Sub: T = \frac{2 \times 30 \times \sin(45^\circ)}{9.8}

Calculated:4.33 s

Step 5[m]

Horizontal Range (R)

R = \frac{v_0^2 \sin(2\theta)}{g}

Sub: R = \frac{(30)^2 \times \sin(2 \times 45^\circ)}{9.8} = \frac{900 \times 1}{9.8}

Calculated:91.84 m

Step 6[J]

Conservation of Mechanical Energy

E_{total} = KE + PE = \frac{1}{2}m v_0^2 + m g h_0

Sub: E_{total} = \frac{1}{2}(1)(30)^2 + (1)(9.8)(0) = 450 + 0

Calculated:450 J

Exam Key Takeaways & Common Sindh Board Questions

Maximum Range Condition: Maximum horizontal range is achieved when launch angle θ = 45° (since sin(2θ) = sin(90°) = 1).
Complementary Angles: Two projection angles that add up to 90° (e.g. 30° and 60° or 20° and 70°) yield the exact same horizontal range for the same initial velocity.
Velocity at Apex: At the highest point, vertical velocity vy = 0, so the velocity is purely horizontal: v_top = vx = v₀ cosθ.
Acceleration is Constant: Acceleration is strictly downward (a_x = 0, a_y = -g) throughout the flight, even at the highest point.

Board Practice & Numerical Challenges

Solve past paper exam numericals and verify them live in the simulation.

Maximum Range Condition

Target question from: Karachi Board 2022 / 2024

A cricket ball is hit with a velocity of 28 m/s at an angle of 45° from ground level (h=0) on Earth. Calculate the maximum horizontal range R achieved.

v₀ = 28 m/sθ = 45°h₀ = 0 mg = 9.8 m/s²

Syllabus Mapping & Learning Objectives

This interactive tool is modeled directly after the official Pakistani textbook curriculam (Sindh Textbook Board, Federal Board Islamabad, and Punjab Textbook Board). It provides live parameter tuning so students can test hypotheses, predict exam outcomes, and observe physical/chemical laws mathematically.

Key Topics Covered

KinematicsVectorsMotion in 2D

More Study Resources

Pair this simulation with verified solved board past papers and chapter revision notes.

Solved Board Past Papers Chapter-Wise Revision Guides Online Scientific Calculator