Stopping distance of a moving vehicle is directly proportional to the square of the initial velocity. This means that if the initial velocity of a vehicle is doubled, the stopping distance will be four times greater. This relationship can be explained by considering the factors that contribute to stopping distance.

The stopping distance of a vehicle is the sum of two components: the thinking distance and the braking distance.

Thinking distance is the distance traveled by the vehicle in the time it takes for the driver to perceive a hazard and react. It depends on various factors such as the driver's reaction time, alertness, and distractions. However, it is not directly related to the speed of the vehicle.

On the other hand, braking distance is the distance traveled by the vehicle while the brakes are applied until it comes to a complete stop. This component is directly influenced by the speed of the vehicle.

The relationship between stopping distance and initial velocity can be explained by the physics of motion. When the brakes are applied, the vehicle decelerates. The deceleration is directly proportional to the initial velocity. This means that the greater the initial velocity, the greater the deceleration required to bring the vehicle to a stop in a given time.

Since deceleration is the rate of change of velocity, a greater deceleration results in a greater change in velocity within a given time interval. Consequently, a greater change in velocity requires a greater distance to come to a stop. As a result, the braking distance increases with the square of the initial velocity.

In mathematical terms, we can express this relationship as:

Stopping distance ∝ (initial velocity)^2

This relationship highlights the importance of considering the speed of a vehicle when it comes to braking and stopping distances. It emphasizes the need for drivers to maintain safe speeds and allow for sufficient braking distances to avoid accidents and ensure the safety of themselves and others on the road.