Question

Why can't displacement -time curve can take sharp turns?

Answer 1

That would result in a discontinuous velocity.

The slope (derivative) of a displacement vs time graph gives the velocity. Velocity must also be continuous, and have a continuous derivative/no "sharp turns" in the line.

Let's think about driving a car. When you step sharply onto the gas petal, you don't go instantly from 25 to 60. You accelerate through all the speeds in between (although, you may increase or decrease the rate at which you go through them). This is an example of a "continuous" function.

Since velocity is continuous, that means that the displacement doesn't have any "sharp turns" in it.

A little more with a picture.



Let's say that you had a graph of a displacement function that looked like this. Is this possible?

Well, no, it's not. The slope goes sharply from 1 to negative 1. That means that the velocity went from 1 to -1. It didn't go through the numbers in between, it just switched. This means that a car was going one direction, then immediately turned around the other direction. It didn't slow down to do this. It didn't stop. It just instantaneously changed direction.

This can't happen in real life. To turn around, you first must slow down, then accelerate in the other direction. So, displacement vs time cannot have sharp turns.