How to Solve a Circular Motion Problem: Net Acceleration and Angle Made with Velocity

Circular motion combines tangential and radial accelerations, requiring both calculations and vector analysis. Let’s break down the solution to a problem involving a car on a circular track.

Problem Overview

A car moves on a circular track of radius 30 meters. Starting from rest, it accelerates tangentially at 0.5 m/s². After 15 seconds, we calculate:

1. The magnitude of its net acceleration.
2. The angle the net acceleration vector makes with the car’s velocity.

Step 1: Tangential Velocity

The tangential velocity (speed along the curve) is calculated using the formula:

velocity = tangential acceleration × time

Since the initial velocity is zero, the result after 15 seconds is:

velocity = 0.5 × 15 = 7.50 m/s

Step 2: Radial (Centripetal) Acceleration

Radial acceleration keeps the car moving along the curve and is calculated as:

radial acceleration = (velocity × velocity) / radius

Substituting values:

radial acceleration = (7.50 × 7.50) / 30 = 1.87 m/s²

Step 3: Net Acceleration

The net acceleration combines tangential and radial accelerations. Since these accelerations are perpendicular, we use the Pythagorean theorem:

net acceleration = square root of (tangential acceleration squared + radial acceleration squared)

Substituting values:

net acceleration = square root of (0.5² + 1.87²) = 1.94 m/s²

Step 4: Angle Between Net Acceleration and Velocity

The angle is calculated using the formula:

tan(angle) = radial acceleration / tangential acceleration

Substituting values:

tan(angle) = 1.87 / 0.5

angle = arctan(1.87 / 0.5) = 75.1°

Key Insights:

• Tangential acceleration drives speed increases along the curve.
• Radial acceleration ensures the car stays on the circular path.
• The net acceleration combines these components to reflect the total effect.
• The angle, 75.1°, highlights that the net acceleration vector points significantly inward due to the stronger radial acceleration.