
Educational Goals
- Investigate the relationship between the horizontal range () of a projectile and its initial velocity () when launched horizontally.
- Apply kinematic equations to predict and verify the proportionality Δx∝vx.
Application of Kinematic Principles
- Calculate initial velocity using vx=Δx sensor/Δt, where Δx sensor is the distance between photodiodes and Δt is the time interval measured.
- Derive the theoretical relationship Δx=vx √(2h/g), where h is the drop height and g is gravitational acceleration.
Experimental Design and Data Analysis
- Use photogate timers and rulers to measure time intervals, sensor distances, and projectile ranges.
- Plot Δx vs. vx to confirm linear proportionality and calculate the constant k=√(2h/g).
Critical Evaluation of Errors
- Identify systematic errors (e.g., rail friction, air resistance) and random errors (e.g., measurement uncertainties in rulers and timers).
Real-World Applications
- Relate findings to engineering and sports scenarios, such as ballistics or javelin throw trajectories.
Collaborative Learning
- Work in teams to compile data, compare results, and refine experimental techniques.
Protocol
- Attach the 2 sensors of the stopwatch to the back of the rail.
- Using the ruler, measure the height of the rail as well as the distance between the sensors.
- Launch the metal ball along the rail using the ball launcher. Observe its fall and measure the distance from the impact point to the ground.
- The time interval measured by the stopwatch is recorded in the results table.
- The range of the marble is recorded in the results table.
- Repeat steps 2 to 5 seven more times, adjusting the intensity of the marble launcher.
- Calculate the speed of the projectile at the end of its horizontal movement. This is the initial speed of the projectile’s movement.
- With the obtained results, determine the relationship that exists between the range of the projectile Delta x and its initial speed vx.
Anticipated Outcomes
Quantitative Results
- Initial Velocity: Calculated using vx=0.100 m/Δt. Example: For Δt=0.079 s vx=1.3 m/s
- Range vs. Velocity: Linear proportionality confirmed by data (e.g., yields Δx=1.295 m).
- Proportionality Constant: For , .
Qualitative Observations
- Higher launch velocities result in longer horizontal ranges.
- Deviations from the ideal linear trend occur due to friction on the rail and measurement inaccuracies.
Graphical Analysis
- Range vs. Velocity Graph: A straight line through the origin confirms Δx=kvx
- Slope Interpretation: Slope equals k, representing the projectile’s time of flight √(2h/g).
Error Analysis
- Systematic Errors: Rail friction reduces the actual , leading to underestimated ranges.
- Random Errors: ±0.001 s timer precision and ±0.005 m ruler uncertainty affect vx and Δx.
Conceptual Understanding
- Students will articulate that horizontal motion (vx) and vertical free fall () are independent.
- Explain why doubling vx doubles Δx if h remains constant.
Summary of Assignment by Grade Range
Grades 6–8
Focus:
- Introduction to projectile motion and basic measurements.
Tasks:
- Launch the ball and measure its range using rulers.
- Record time intervals from the photogate timer.
- Discuss how launch speed affects distance traveled.
Expected Outcomes:
- Recognize that faster launches result in longer ranges.
- Practice tabulating Δt, vx, and Δx.
- Identify simple error sources (e.g., inconsistent launches).
Grades 9–10
Focus:
- Quantitative analysis of kinematics.
Tasks:
- Calculate vx and plot Δx vs. vx.
- Derive k=√(2h/gand compare it to the graph’s slope.
- Use Δx=vxk to predict ranges for untested velocities.
Expected Outcomes:
- Apply unit conversions (e.g., cm → m, ms → s).
- Explain deviations from the theoretical model using friction and measurement errors.
Grades 11–12
Focus:
- Advanced error analysis and experimental optimization.
Tasks:
- Perform uncertainty propagation for vx and Δx.
- Calculate percent error between experimental and theoretical k.
- Redesign the experiment to minimize rail friction (e.g., lubricated rails).
Expected Outcomes:
- Write lab reports with error bars on graphs and statistical analysis.
- Propose studies on angled launches or variable heights.
- Evaluate the impact of air resistance using high-speed cameras.
Laboratory essentials
Instruments
Electric ball launcher
Sandbox
Photodiodes sensors
Metal ball
50 cm ruler