Projectile Motion Lab

Projectile Motion Lab: Exploring Two-Dimensional Kinematics

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Projectile motion is one of the most elegant examples of physics in action. From a basketball arcing toward the hoop to a water fountain's graceful spray, projectiles follow predictable paths governed by surprisingly simple mathematics. This lab simulation lets you explore these principles firsthand, launching virtual projectiles and analyzing their trajectories in real-time.

What makes projectile motion particularly fascinating is the independence of horizontal and vertical motion. A ball thrown horizontally from a cliff takes exactly the same time to hit the ground as one simply dropped—the horizontal velocity doesn't affect the vertical fall [1]. This counterintuitive fact lies at the heart of understanding two-dimensional kinematics.

The Physics of Projectile Motion

Decomposing the Motion

When a projectile is launched at an angle θ with initial speed v₀, we can break the motion into two independent components:

This decomposition is the key insight: the horizontal motion knows nothing about the vertical motion, and vice versa. Gravity acts only in the vertical direction, leaving horizontal velocity unchanged [2].

Key Equations for Projectile Motion

Position equations:

• x(t) = v₀ cos(θ) · t
• y(t) = h₀ + v₀ sin(θ) · t − ½gt²

Velocity equations:

• vₓ(t) = v₀ cos(θ) = constant
• vᵧ(t) = v₀ sin(θ) − gt

Derived quantities:

• Time of flight: T = (v₀ sin(θ) + √[(v₀ sin(θ))² + 2gh₀]) / g
• Maximum height: H = h₀ + (v₀ sin(θ))² / (2g)
• Range (h₀ = 0): R = v₀² sin(2θ) / g

Learning Objectives

After completing this lab, you should be able to:

1. Predict trajectories given initial conditions (angle, speed, height)
2. Explain why horizontal and vertical motions are independent
3. Calculate range, maximum height, and time of flight
4. Identify the launch angle that maximizes range on level ground
5. Compare complementary angles and explain why they produce equal ranges
6. Analyze how gravity affects only the vertical component of motion

Exploration Activities

Activity 1: Maximum Range Investigation

Objective: Discover the optimal launch angle for maximum range.

Procedure:

1. Set initial speed to 20 m/s and launch height to 0 m
2. Launch at 30° and record the range
3. Repeat for 45°, 60°, and 75°
4. Create a data table and identify the angle giving maximum range
5. Test angles near 45° (e.g., 43°, 47°) to refine your answer

Expected result: Maximum range occurs at 45° when launching from ground level. The formula R = v₀²sin(2θ)/g is maximized when sin(2θ) = 1, which occurs at θ = 45° [3].

Activity 2: Complementary Angle Pairs

Objective: Verify that complementary angles produce equal ranges.

Procedure:

1. Launch at 30° with v₀ = 25 m/s and record the range
2. Launch at 60° with the same speed and compare
3. Test other pairs: 20° & 70°, 35° & 55°
4. Record both range and maximum height for each

Key insight: While ranges are equal, maximum heights differ dramatically. The steeper angle goes higher but takes the same horizontal distance to land.

Activity 3: Height vs. Range Trade-off

Objective: Understand how launch height affects trajectory.

Procedure:

1. Set angle to 45° and v₀ = 20 m/s
2. Launch from h₀ = 0 m and record range and time
3. Increase to h₀ = 10 m, then 20 m, then 30 m
4. Plot how range increases with launch height

Analysis question: Does doubling the launch height double the range?

Activity 4: Planetary Comparison

Objective: Explore how gravity affects projectile motion.

Procedure:

1. Launch a projectile on Earth (g = 9.81 m/s²) at 45°, 20 m/s
2. Switch to Moon gravity (g = 1.62 m/s²) and repeat
3. Try Mars (g = 3.72 m/s²)
4. Calculate the ratio of ranges and compare to the ratio of gravities

Key formula: Since R ∝ 1/g, range on the Moon should be roughly 6× that on Earth for identical launches.

Real-World Applications

Projectile motion principles apply far beyond physics classrooms:

Reference Data: Gravitational Acceleration

Data sourced from NASA planetary fact sheets [4].

Challenge Questions

Level 1 (Basic):

1. A ball is thrown horizontally at 15 m/s from a 20 m cliff. How long does it take to hit the ground?

Level 2 (Intermediate): 2. What two launch angles give a range of 30 m when v₀ = 20 m/s? (Hint: use sin(2θ) = gR/v₀²)

Level 3 (Advanced): 3. A projectile is launched from ground level and passes through a point 10 m horizontally and 5 m vertically at t = 0.5 s. Find v₀ and θ.

Level 4 (Challenge): 4. Derive the equation for the trajectory y(x) eliminating time. Show it's a parabola.

Level 5 (Extension): 5. If air resistance is proportional to velocity (F_drag = -bv), qualitatively describe how the trajectory changes. Why does optimal range angle decrease below 45°?

Common Mistakes to Avoid

Frequently Asked Questions

Q: Why is 45° the optimal angle for maximum range? A: The range formula R = v₀²sin(2θ)/g reaches maximum when sin(2θ) = 1, which occurs at 2θ = 90°, giving θ = 45°. At this angle, the horizontal and vertical velocity components are equal, providing the best balance between flight time and horizontal speed [3].

Q: Does the mass of the projectile affect its trajectory? A: In ideal conditions (no air resistance), mass doesn't affect trajectory. All objects experience the same gravitational acceleration g. This was famously demonstrated when Apollo 15 astronaut David Scott dropped a hammer and feather on the Moon—they fell at the same rate [5].

Q: Why do complementary angles give the same range? A: Because sin(2θ) = sin(180° - 2θ). For angles θ and (90° - θ), we have sin(2θ) = sin(180° - 2θ) = sin(2(90° - θ)), so both produce identical range values [3].

Q: How does air resistance change things? A: Air resistance reduces both range and maximum height. It also breaks the symmetry of the trajectory—the descending path is steeper than the ascending path. The optimal launch angle decreases below 45° with air resistance [6].

Q: What's the fastest way to solve projectile problems? A: Follow this systematic approach: (1) Identify what's given and what's asked, (2) Break v₀ into components, (3) Write separate equations for x and y motion, (4) Use the equation that directly connects your knowns to unknowns, (5) Check that your answer makes physical sense.

References

[1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics, 10th Edition. Wiley. Chapter 4: Motion in Two and Three Dimensions.

[2] Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers, 10th Edition. Cengage Learning. Chapter 4.

[3] Knight, R. D. (2016). Physics for Scientists and Engineers: A Strategic Approach, 4th Edition. Pearson. Section 4.3: Projectile Motion.

[4] NASA. (2024). Planetary Fact Sheets. NASA Goddard Space Flight Center. Available at: https://nssdc.gsfc.nasa.gov/planetary/factsheet/

[5] NASA. (1971). Apollo 15 Hammer-Feather Drop. NASA History Division. Available at: https://nssdc.gsfc.nasa.gov/planetary/lunar/apollo_15_feather_drop.html

[6] Giordano, N. J. (2012). Computational Physics, 2nd Edition. Pearson. Chapter 2: Realistic Projectile Motion.

About the Data

Gravitational acceleration values are sourced from NASA's planetary fact sheets. The physics equations presented follow standard notation from university physics textbooks. All calculations in the simulation use exact values to ensure accuracy within floating-point precision.

Citation Guide

To cite this simulation in academic work:

Simulations4All. (2025). Projectile Motion Lab [Interactive simulation]. Simulations4All Educational Platform. Retrieved from https://simulations4all.com/simulations/projectile-motion-lab

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