Launch Analysis

Physics Explanation: Integration Method

Velocity Formula: v = v₀ + ∫a·dt (integration of acceleration over time)

Height Formula: h = h₀ + ∫v·dt (integration of velocity over time)

How it works: We use calculus to integrate the accelerometer data. At each time step, we:

1. Read the acceleration from the sensor (subtract 1g to remove Earth's gravity)
2. Calculate the time elapsed since the last reading (dt)
3. Add acceleration × dt to the velocity (v = v + a·dt)
4. Add velocity × dt to the height (h = h + v·dt)

Results: Maximum velocity reached was 23.1 m/s, and maximum height was 75.6 meters.

Reading the Graph

The graph shows total acceleration (in g-force) over time. Peaks indicate moments of high force, such as:

• Launch: Initial spike when catapult releases
• Flight: Lower readings during free-fall (~1g)
• Impact: Large spike when pumpkin hits the ground

Understanding the Flight Path

This graph shows the parabolic trajectory of the pumpkin. The line is color-coded to show flight phases:

• Green (Rising): Pumpkin rises after launch, velocity > 0.5 m/s upward
• Gold (Apex): Near maximum height, velocity between -0.5 and 0.5 m/s
• Red (Falling): Pumpkin descending, velocity < -0.5 m/s (negative = downward)

Detection Method: We determine when the pumpkin is falling by tracking the integrated velocity. When velocity becomes negative (meaning downward motion), the pumpkin has passed its peak and is falling back to Earth. This is an educated guess based on the velocity calculations from the accelerometer data.

In projectile motion, this creates a parabola. The shape is determined by initial velocity and launch angle. Hover over the graph to see exact values and flight phase at any point!

Velocity Analysis

The velocity graph shows how the pumpkin's speed changes throughout flight:

• Launch spike: Rapid acceleration as the catapult releases
• Deceleration: Velocity decreases as gravity slows the upward motion
• Zero at peak: Velocity reaches zero at maximum height
• Negative velocity: Indicates downward motion during descent

Velocity Calculation in Miles Per Hour

Conversion Formula:

v_mph = v_m/s × 2.23694

How it works: We calculate velocity in meters per second (m/s) using the integration method described above (v = ∫a·dt). To convert to miles per hour, we multiply by the conversion factor 2.23694.

Example: If the pumpkin reaches a maximum velocity of 23.1 m/s, this equals 51.6 mph (23.1 × 2.23694 = 51.6).

Physical Interpretation: The change in velocity over time (Δv/Δt) is acceleration. By integrating the accelerometer data, we're summing up all the tiny velocity changes over each time step to get the total velocity at any moment.

Rotational Motion

The gyroscope measures rotation in degrees per second (dps):

• Launch spin: Initial rotation imparted by the catapult release
• Tumbling: Irregular rotation during flight as the pumpkin tumbles
• Impact spike: Sudden rotation changes when hitting the ground