Mechanics & Simple Machines · AP Physics 1 Unit 1 (2D Kinematics)
Projectile & Trajectory Motion Simulator
Key exam takeaway: Horizontal and vertical motions in projectile motion are completely independent and share the exact same flight time.
- Client-side
- No Sign-up
- Live Derivation
- Projector Mode
- Print Worksheet
3D Simulation Stage● 60 FPS
Parameter Controls
m/s
°
m
m/s²
Live Derivations & Metrics
Flight Time t
--
Max Apex Height H
--
Horizontal Range X
--
Impact Velocity
--
Real-Time Formula Derivation
Governing Formula:
--Substitution:
--Regime: --
Worked Examples & Exam Question Deep Dive
Worked Problem
Problem: A rock is thrown horizontally from a 20m cliff at 15m/s. Find flight time and horizontal range. (g=10m/s²)
Step-by-Step Derivation:
- Vertical fall: h = 0.5gt² => t = sqrt(40/10) = 2s. Horizontal range: X = vx*t = 15*2 = 30m.
Worked Problem
Problem: A ball is kicked at 20m/s at 30° to horizontal. Find max apex height and total range. (g=10m/s²)
Step-by-Step Derivation:
- vy = 20*sin30° = 10m/s, vx = 10sqrt(3) m/s. Apex height H = 10²/(2*10) = 5m. Flight time t = 2s. Range X = 20sqrt(3) ≈ 34.64m.
Student Inquiry Worksheet & Practice Problems
Q1. Variant 1: What is the constant ratio between velocity angle tangent tanθ and displacement angle tangent tanα?
Answer Key: tanθ = gt/v0 and tanα = 0.5gt/v0. Thus tanθ = 2 tanα, meaning velocity angle tangent is strictly twice displacement angle tangent.
Q2. Variant 2: When launched from an elevated cliff (h > 0), is the optimal angle for maximum range still 45°?
Answer Key: Less than 45°. The extra vertical drop extends flight time, so prioritizing horizontal speed yields greater range.
Q3. Variant 3: Why is a level bomber directly above its dropped bomb at detonation neglecting drag?
Answer Key: Both the airplane and the bomb maintain the exact same horizontal velocity by inertia.
Frequently Asked Questions & Misconceptions
What does the dual-ball drop experiment prove?
It proves that vertical motion in projectile trajectory is identical to free fall and completely independent of horizontal speed.
How does air resistance alter real-world trajectories?
Air drag reduces range, steepens descent trajectory, and lowers the optimal launch angle from 45° to approximately 38°-42°.
Is projectile motion uniformly accelerated motion?
Yes! Because gravity is constant, acceleration is constant (a = g downward), making it uniformly accelerated curvilinear motion.
Will a dart aimed directly at a monkey hit if the monkey drops at launch?
Yes, always (before hitting ground)! Both dart and monkey accelerate downward at g, maintaining the line of sight.