Lecture Notes: Car Motion on a Hill

Introduction

• Scenario: A car is at the top of a hill and runs out of gas.
• Objective: Determine the velocity or kinetic energy needed for the car to reach a gas station on another hill.
• Assumptions:
  • System is under a conservative force.
  • Total mechanical energy is conserved (similar to a gravitational field).
  • No friction force is present.

Problem Analysis

• Initial condition: Car at rest at the top of a hill.
• Maximum height without additional speed is equal to its starting height due to energy conservation.
• Potential energy converts to kinetic and back to potential energy as the car moves.
• Additional speed is necessary for the car to reach higher gas station.

Energy Conservation Principle

• Initial and final energy states must be equal:
  • Total Energy at Point 1 (Initial) = Total Energy at Point 2 (Final)
• Components:
  • Kinetic Energy (KE): KE = 1/2 * m * v^2
  • Potential Energy (PE): ( mgh )

Calculations

• Hill height: 20 meters.
• Gas station height: 30 meters.
• Energy equations:
  • Initial Total Energy (Point 1): KE + PE
  • Final Total Energy (Point 2): PE only (as car just reaches the gas station with zero speed)
• Simplified Equation:
  • Mass cancels out from both sides.
  • Solve for minimum speed: (𝑣𝑚𝑖𝑛=𝑠𝑞𝑟𝑡2𝑔(ℎ2−ℎ1))
  • Given: g = 9.8 ext m/s^2 , (h_1 = 20 ext m ), ( h_2 = 30 ext m)
  • Result: Minimum speed (v_min = 14 ext m/s )

Follow-Up Question

• Calculate the speed of the car at a height of 25 meters.
• Encouraged to submit answers and questions in the comment section.

Conclusion

• Minimum speed required: 14 m/s to reach the gas station.
• Reiterated the absence of friction in the calculation.
• Invitation for engagement: Questions and comments are welcomed.
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