Overview
This lecture covers Lab 3 on projectile motion, focusing on calculating initial velocity and predicting the horizontal distance of a projectile using a spring-loaded cannon.
Projectile Motion Basics
- Projectile motion involves an object launched with initial velocity (v_0) at an angle (\theta), moving under gravity.
- The launch point is chosen as the origin for calculations ((x_0 = 0), (y_0 = 0)).
- The velocity components are: (v_{0x} = v_0 \cos \theta), (v_{0y} = v_0 \sin \theta).
- Acceleration in the x-direction ((a_x)) is zero; in the y-direction ((a_y)) is (-9.8, \mathrm{m/s^2}).
Key Equations
- Horizontal position: (x = v_0 \cos \theta \cdot t)
- Vertical position: (y = v_0 \sin \theta \cdot t - 4.9t^2)
- For horizontal launch ((\theta = 0)), (y) equation simplifies as (v_0 \sin 0 = 0).
Step 1: Finding Initial Velocity (Horizontal Launch)
- Set cannon angle to (0^\circ) (horizontal).
- Measure vertical drop ((y = -1.163, \mathrm{m})) and horizontal range ((x = 3.294, \mathrm{m})).
- Use the vertical position equation to solve for time ((t)), then the horizontal equation to solve for initial velocity ((v_0)).
- Remember to use negative (y) value for falling below the origin.
Step 2: Predicting Range at an Angle
- Set cannon to fire at (59^\circ).
- Vertical drop stays (y = -1.163, \mathrm{m}); initial velocity (v_0) is assumed the same as step 1.
- Use the vertical equation to solve for time ((t)); select the positive root from the quadratic equation.
- Substitute (t) into the horizontal equation to predict range ((x)).
- Actual measured range may be slightly shorter due to speed differences at different angles.
Key Terms & Definitions
- Projectile — An object moving under the influence of gravity only after initial launch.
- Initial velocity ((v_0)) — The speed at which the projectile leaves the cannon.
- Angle ((\theta)) — The direction above horizontal at which the projectile is launched.
- Origin — The reference starting point for measurements ((x_0 = 0), (y_0 = 0)).
- Quadratic equation — An equation involving (t^2), which may yield two solutions for time; use the positive one.
Action Items / Next Steps
- Calculate time and initial velocity from the horizontal launch data.
- Calculate the predicted horizontal distance using the angle and found initial velocity.
- Submit calculations for both initial speed and predicted range at (59^\circ).
- Review solving quadratic equations for projectile motion scenarios.