Rocket Run Lesson Plan Rocket Run
Lesson Plan | Honors Physics & Calculus
Mission Brief
Learning Objective
Students will interpret the derivative as instantaneous velocity in projectile motion. They will use the limit definition to solve for velocity, find maximum height (where \(v=0\)), and calculate impact velocity.
Materials Needed
Projector & Video Link
Rocket Run Activity Sheets
Scientific Calculators
Flight Procedure
5 min Warm-up
Speedometer vs. Average
Engage the class with a discussion: "What is the difference between your average speed on a 3-hour road trip vs. the speed shown on your speedometer at any given second?"
Key Insight: Average speed is \(\Delta d / \Delta t\) over a long interval; the speedometer is the limit as \(\Delta t \to 0\).
15 min Viewing
Visualizing the Launch
Watch the rocket section (6:27-10:48) of the Introduction to Derivatives video.
Pause at 9:08 to have students predict the sign of the velocity at t=11.
Discussion: Why is the velocity negative when the rocket is falling? What does a derivative of zero represent physically?
20 min Main
Rocket Launch Simulation
Distribute the activity sheets. Students work in groups to analyze unique flight functions.
Task 1: Use the limit definition \(\lim_{h \to 0} \frac{h(t+h)-h(t)}{h}\) to find the velocity function \(v(t)\).
Task 2: Solve for Peak Altitude (\(v=t=0\)) and Impact Velocity (\(h=0\)).
5 min Debrief
Mission Debrief
Compare impact velocities across groups. Why do rockets with higher initial coefficients hit the ground harder?
Exit Ticket: Write one sentence explaining the relationship between the slope of a height graph and the velocity of the rocket.
Common Misconceptions
• Students may think negative velocity means "slowing down" (it means direction).
• Forgetting to distribute the negative sign when expanding \(h(t+h) - h(t)\).
• Confusing \(t=0\) (launch) with \(v=0\) (peak).
Extension for Honors
Ask students to derive the generic velocity function for \(h(t) = v_0t - \frac{1}{2}gt^2\). Can they show that the time to peak is always exactly half the total flight time?
Rocket Run Activity Sheet Flight Data Log
Operation: Instantaneous Velocity Simulation
Name:
Date:
STATUS: [CALCULATION MODE]
VARIABLE: t = time (s)
FUNCTION: h(t) = height (m)
Assigned Rocket Function
Your group has been assigned the following height function for a test rocket. Use the limit definition of the derivative to analyze its flight.
\[h(t) = 80t - 5t^2\]
1
Calculate the Velocity Function \(v(t)\)
Use the limit definition: \(v(t) = \lim_{h \to 0} \frac{h(t+h) - h(t)}{h}\). Show all algebraic expansions below.
2
Finding Peak Altitude
At the peak, velocity is zero. Solve for \(t\) when \(v(t) = 0\), then find the height at that time.
Max Height (m):
3
Impact Analysis
Calculate the time of impact (\(h(t) = 0\)), then find the velocity \(v(t)\) at that specific moment.
Impact Velocity (m/s):
Post-Flight Analysis
1. Why is the velocity positive during launch and negative at impact?
2. Compare your impact velocity with another group. If their rocket had an initial term of \(120t\) instead of \(80t\), what happened to their impact velocity? Why?
Flight Code: RKT-DERIV-11 Physics Department | Calculus Section 4.2 Checksum: 0x51A23
Rocket Run Slides ROCKET RUN
Calculating Instantaneous Velocity through Derivatives
Physics Projectile Motion
Calculus Rate of Change
Speedometer vs. Average
WARM-UP
"What is the difference between your average speed on a road trip vs. the speed shown on your speedometer at any given second?"
Average Rate
The total distance divided by the total time over a long interval.
Instantaneous Rate
The rate of change at one exact moment in time.
Visualizing the Rocket
VIDEO ANALYSIS
Embedded media
Section: Rocket Launch Scenario (6:27 - 10:48)
Key Questions
What does the sign of the velocity tell us about the rocket's direction?
Why is the derivative at the peak exactly zero?
What happens to height when velocity is negative?
The Math Foundation
Instantaneous Velocity Formula
\[v(t) = \lim_{h \to 0} \frac{h(t+h) - h(t)}{h}\]
\(h\)
Difference in time intervals
\(\lim_{h \to 0}\)
Shrinking the interval to zero
Slope
Rate of height change
Mission Parameters
Instructions
Phase 1: Derivation
Use the limit definition to find your rocket's velocity function \(v(t)\). Show all work.
Phase 2: Peak
Solve for \(v(t) = 0\) to find the time and altitude of the rocket's highest point.
Phase 3: Impact
Solve for \(h(t) = 0\) to find when it hits the ground, then find the impact velocity.
"Check your coefficients! A small error in expansion leads to a massive crash in velocity."
Rocket Run Answer Key Answer Key
Rocket Run Activity Simulation
Confidential
Standard Rocket Function: \(h(t) = 80t - 5t^2\)
1. Derivative Derivation
\[v(t) = \lim_{h \to 0} \frac{[80(t+h) - 5(t+h)^2] - [80t - 5t^2]}{h}\]
\[v(t) = \lim_{h \to 0} \frac{80t + 80h - 5(t^2 + 2th + h^2) - 80t + 5t^2}{h}\]
\[v(t) = \lim_{h \to 0} \frac{80h - 10th - 5h^2}{h}\]
\[v(t) = \lim_{h \to 0} (80 - 10t - 5h)\]
\[v(t) = 80 - 10t\]
2. Peak Altitude Analysis
Time at peak: \(80 - 10t = 0 \implies t = 8 \text{ s}\)
Height at peak: \(h(8) = 80(8) - 5(8^2)\)
\(h(8) = 640 - 320\)
Max Height = 320 m
3. Impact Analysis
Time of impact: \(80t - 5t^2 = 0 \implies 5t(16 - t) = 0\)
\(t = 16 \text{ s}\) (Discard \(t=0\))
Velocity at impact: \(v(16) = 80 - 10(16)\)
\(v(16) = 80 - 160\)
Impact Velocity = -80 m/s
Other Rocket Variations
Function \(h(t)\) Velocity \(v(t)\) Peak Time Peak Height Impact Velocity \(120t - 5t^2\) \(120 - 10t\) 12s 720m -120m/s \(60t - 3t^2\) \(60 - 6t\) 10s 300m -60m/s \(100t - 4.9t^2\) \(100 - 9.8t\) 10.2s 510.2m -100m/s
Post-Flight Reflection Notes
1. Sign Conventions
Positive velocity indicates upward movement (height increasing). Negative velocity indicates downward movement (height decreasing). At the peak, the slope of the height graph is horizontal, thus \(v(t) = 0\).
2. Symmetry Observation
In all parabolic cases without air resistance, the impact velocity is exactly equal in magnitude and opposite in sign to the initial velocity (\(v_0\)). The peak time is exactly half the flight time.
