Overview
This lecture demonstrates how to solve a physics problem involving a sandbag dropped from a rising balloon, focusing on equations of motion with constant acceleration.
Problem Setup
- A balloon ascends vertically at 6 m/s.
- A sandbag is dropped from the balloon with an initial upward velocity of 6 m/s at ( t = 0 ).
- The sandbag hits the ground after 8 seconds.
- Required: speed of the sandbag at impact and altitude of the balloon at that instant.
Key Equations and Steps
- Use constant acceleration motion equations to solve for positions and velocity.
- Define upwards as the positive direction.
- Acceleration due to gravity is ( g = -9.81 ) m/s² (downwards).
Calculating Sandbag Position (when it hits ground)
- Use ( s = s_0 + v_0 t + \frac{1}{2} a t^2 ).
- ( s_0 = 0 ), ( v_0 = 6 ) m/s, ( a = -9.81 ) m/s², ( t = 8 ) s.
- Substitute: ( s = 0 + 6(8) + 0.5(-9.81)(8^2) ).
- Result: ( s_{\text{bag}} = -265.9 ) m (downward, indicating it is below the release point).
Calculating Balloon Position (after 8 s)
- Balloon moves at constant velocity, so ( a = 0 ).
- ( s = s_0 + v_0 t ), ( s_0 = 0 ), ( v_0 = 6 ) m/s, ( t = 8 ) s.
- ( s_{\text{balloon}} = 6 \times 8 = 48 ) m (above release point).
Finding Balloon Altitude (above ground)
- Altitude is the vertical distance between the balloon and ground at ( t = 8 ) s.
- ( \text{Altitude} = s_{\text{balloon}} - s_{\text{bag}} = 48 - (-265.9) = 313.9 \approx 314 ) m.
Speed of Sandbag at Impact
- Use ( v = v_0 + a t ).
- ( v_0 = 6 ) m/s, ( a = -9.81 ) m/s², ( t = 8 ) s.
- ( v = 6 + (-9.81)(8) = -72.5 ) m/s (downwards).
- Speed (magnitude) is ( 72.5 ) m/s.
Key Terms & Definitions
- Constant Acceleration — Motion with unchanging acceleration, such as gravity near Earth's surface.
- Initial Velocity (( v_0 )) — The sandbag's velocity at the moment of release.
- Altitude — The vertical distance from the ground to the balloon.
- g (Gravity) — Acceleration due to Earth's gravity, approximately ( 9.81 ) m/s² downwards.
Action Items / Next Steps
- Practice applying equations of motion to problems with constant acceleration.
- Review the signs (positive/negative) for direction in kinematics problems.