088 – The relationship between the resultant force and acceleration

Educational Goals

• Investigate the relationship between the horizontal range ($\Delta x$) of a projectile and its initial velocity (vx​) 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

1. Attach the 2 sensors of the stopwatch to the back of the rail.
2. Using the ruler, measure the height of the rail as well as the distance between the sensors.
3. 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.
4. The time interval measured by the stopwatch is recorded in the results table.
5. The range of the marble is recorded in the results table.
6. Repeat steps 2 to 5 seven more times, adjusting the intensity of the marble launcher.
7. Calculate the speed of the projectile at the end of its horizontal movement. This is the initial speed of the projectile’s movement.
8. 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 $,$ $.$
• Range vs. Velocity: Linear proportionality confirmed by data (e.g., vx​=2.4m/s yields Δx=1.295 m).
• Proportionality Constant: For $h=1.406\text{m}$, k=√(2×1.406/9.8)$\approx 0.54\text{s}$.

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 $.$
• Slope Interpretation: Slope equals k, representing the projectile’s time of flight √(2h/g).

Error Analysis

• Systematic Errors: Rail friction reduces the actual vx​, 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 (h=0.5​gt²) 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/g) and 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

Products