How to Find Velocity from Acceleration and Radius?

You can find the velocity from centripetal acceleration and radius by rearranging the formula for centripetal acceleration.

When an object moves in a circle at a constant speed, it experiences centripetal acceleration directed towards the center of the circle. This specific type of acceleration is related to the object's tangential velocity (speed) and the radius of the circular path.

The relationship between centripetal acceleration (a c), tangential velocity (v), and the radius of the circular path (r) is given by the formula provided in the reference:

a c = v 2 r

To find the velocity (v), you need to solve this equation for v.

Steps to Find Velocity

Follow these steps to determine the tangential velocity using centripetal acceleration and radius:

1. Identify Values and Convert Units:

   • Determine the value of the radius (r) of the circular path.
   • Determine the value of the centripetal acceleration (a c).
   • As per the reference, convert these values to standard SI units: radius in meters (m) and centripetal acceleration in meters per second squared (m/s²). If your values are in other units (like kilometers, centimeters, km/h, g's), convert them first.

2. Rearrange the Formula:

   • Start with the formula: a c = v 2 r
   • Multiply both sides of the equation by r to isolate v²:
     a c * r = (v 2 / r) * r
a c * r = v 2
   • Take the square root of both sides to solve for v:
     sqrt(a c * r) = sqrt(v 2)
v = sqrt(a c * r)

3. Calculate the Velocity:

   • Plug the values for a c (in m/s²) and r (in m) into the rearranged formula v = sqrt(a c * r).
   • Calculate the result. The velocity v will be in meters per second (m/s).

Summary of Formula and Calculation

Here's a quick look at the formula rearrangement:

Quantity	Symbol	Unit (SI)	Formula
Centripetal Acceleration	a c	m/s²	a c = v 2 / r
Tangential Velocity	v	m/s	**v = sqrt(a c * r)**
Radius	r	m

Practical Example

Let's say an object is moving in a circle with a radius of 5 meters and experiences a centripetal acceleration of 20 m/s². What is its tangential velocity?

1. Identify Values and Convert:
   • r = 5 m (already in meters)
   • a c = 20 m/s² (already in m/s²)
2. Use the Formula:
   • v = sqrt(a c * r)
   • v = sqrt(20 m/s² * 5 m)
3. Calculate:
   • v = sqrt(100 m²/s²)
   • v = 10 m/s

So, the tangential velocity of the object is 10 m/s.

This method is crucial in physics and engineering for understanding and calculating the speed of objects in circular motion, from satellites orbiting Earth to cars turning corners.