## Maths in a Minute: The wagon wheel effect

One of the greatest, most significant and major innovations in science is the wheel. However, most of us have overlooked a fascinating observation in a moving wheel. Sometimes, the wheels of an automobile are observed to be rotating backwards in movies or advertisements, even though the vehicle is moving forwards. Why? Let's explore the reason behind this intriguing phenomenon!

This phenomenon is usually observed in high-speed vehicles. The perceived direction of motion is influenced by both the speed of the wheel as well as the video camera's capturing speed. Through diagrams and mathematics, we can illustrate the effect of these two factors on the observed direction of motion.

We have created two cases which depict the observed motion of the wheel in both clockwise and anticlockwise direction. In these cases, we will change the wheel's speed and keep the camera's capturing speed the same. We discover that with low speeds, the wheel is noticed rotating in the clockwise direction while in the high-speed case it is perceived to be rotating in the anticlockwise direction.

For the wheel of a slow-moving car, shown below, we start with the wheel at a bearing of 0°. On each observation, each frame captured by the camera, it moves 30° clockwise as represented in the diagram. Thus, we observe the wheel moving clockwise.

The wheel of a slow moving car, captured by a video camera, appears to be moving clockwise.

For the wheel of a fast-moving car, shown below, we again start with the wheel at a bearing of 0°. But now, with each observation captured by the camera, the wheel moves 300° clockwise. This movement of 300° clockwise can also be thought as moving 60° anti-clockwise. Our brain always interprets motion as that with the shortest angle. Thus, even though the wheel moves clockwise, we observe anti-clockwise motion, that is, the wheel moving in reverse.

The wheel of a fast moving car, captured by a video camera, appears to be moving anticlockwise.

This optical illusion of perceiving the motion of a wheel moving in a backwards direction, when in fact the vehicle is moving forwards, is known as *wagon-wheel* or *stage-coach effect*.

A simple model can be constructed to observe the wagon-wheel effect using a speed-controlled motor and a mobile phone, as shown in the video below.

### About the author

Aadit Jain is high-school student with keen interest in maths, physics and their combined effect. He is always curious about how things work.