Vehicle's Angular Rate, Angle, Range Calculator

What is Angular Rate?

Angular rate, also known as angular velocity, is a measure of how quickly an object rotates about an axis. In the context of vehicle motion, it describes how fast a vehicle is turning. The angular rate is typically expressed in radians per second (rad/s) and is crucial for understanding the dynamics of rotating objects or vehicles making turns.

Formulas

The key formulas for this calculator are:

1. Angular Rate: \[ \omega = \frac{v}{r} \]
2. Angle: \[ \theta = \omega t \]
3. Range: \[ s = vt \]

Where:

• \( \omega \) is the angular rate (rad/s)
• \( v \) is the velocity (m/s)
• \( r \) is the radius of turn (m)
• \( \theta \) is the angle (rad)
• \( t \) is the time (s)
• \( s \) is the range or distance traveled (m)

Calculation Steps

Let's calculate these values for a vehicle with the following parameters:

• Velocity (\( v \)) = 20 m/s
• Radius of turn (\( r \)) = 100 m
• Time (\( t \)) = 10 s

1. Calculate the angular rate: \[ \omega = \frac{v}{r} = \frac{20 \text{ m/s}}{100 \text{ m}} = 0.2 \text{ rad/s} \]
2. Calculate the angle: \[ \theta = \omega t = 0.2 \text{ rad/s} \times 10 \text{ s} = 2 \text{ rad} \]
3. Calculate the range: \[ s = vt = 20 \text{ m/s} \times 10 \text{ s} = 200 \text{ m} \]

Example and Visual Representation

Let's visualize the vehicle's motion:

This diagram illustrates:

• The circular path of the vehicle (blue circle)
• The radius of turn (\( r \)) (red line)
• The range or distance traveled (\( s \)) (green arc)
• The angle (\( \theta \)) swept by the vehicle
• The velocity (\( v \)) of the vehicle (yellow arrow)