Analyzing Car's Acceleration on a Circular Path

Problem Overview

A stationary car accelerates at 0.5 meters per second squared for 15 seconds in a circular path with a radius of 30 meters. We need to determine the net acceleration and its direction relative to the tangential velocity component after 15 seconds.

Key Concepts

1. Linear and Radial Acceleration: Understanding the components of acceleration in circular motion.
2. Net Acceleration: Calculating the vector sum of radial and tangential accelerations.
3. Direction of Net Acceleration: Determining the angle between net acceleration and tangential velocity.

Step-by-Step Solution

Part 1: Determining Radial and Tangential Accelerations

1. Tangential Acceleration (AT): Given as 0.5 m/s².
2. Linear Velocity After 15 Seconds: Final velocity = Initial velocity + Acceleration × Time = 0 + 0.5 × 15 = 7.5 m/s.
3. Radial Acceleration (AR): AR = V² / radius = (7.5)² / 30 = 1.875 m/s².

Part 2: Calculating Net Acceleration

1. Net Acceleration (AN): AN = √(AR² + AT²) = √(1.875² + 0.5²) ≈ 1.94 m/s².

• • Angle of Net Acceleration:Use tan θ = Opposite / Adjacent = AR / AT.
  • tan θ = 1.875 / 0.5.
  • θ = tan⁻¹(1.875 / 0.5) ≈ 75.1 degrees.

Conclusion

• After 15 seconds, the car has a net acceleration of approximately 1.94 m/s².
• The direction of this net acceleration makes an angle of approximately 75.1 degrees with the tangential velocity component, indicating the combined effect of linear and radial accelerations in circular motion.