# Comparing u-dot and Acceleration: What's the Difference?

• SpaceTrekkie
In summary, u-dot is the time derivative of speed, while a is the time derivative of velocity. These two quantities are not necessarily the same, as seen in the example of uniform circular motion. The dot product u dot a can be rewritten as u*u-dot, where u-dot represents the change in speed over time. This can be further simplified using the formula x(dx/dt) = (1/2)(d/dt)x^2.
SpaceTrekkie

## Homework Statement

The problem says to show that the dot product u dot a = u*u-dot where u-dot is the differentiation with respect to time.

How is u-dot different from just the acceleration?

My teacher said that "a is the magnitude of the 3-acceleration.

u-dot is the time derivative of the speed.

In the first case, you first differentiate the velocity with time and then take the magnitude. In the second case, you first take the magnitude and then differentiate with time. These are not necessarily the same. For example, consider uniform circular motion. In that case, u-dot is always zero, but a is never zero."

but I am not sure I understand this...

N/A

## The Attempt at a Solution

No idea where to begin. I know that if u and a are perpendicular them the dot product is 0, but that is about it.

Okay, I figured out that u-dot is the change in SPEED over time, while a = the change in VELOCITY over time. So the u-dot has no direction...but where to go from there, is still a mystery to me...

Firstly, your topic name is a bit of a misnomer, isn't it? This doesn't have anything to do with either SR or 4-vectors, but perhaps you didn't know that.

Anyways here's a hint : we can write $$x \frac{dx}{dt} = \frac{1}{2} \frac {d}{dt} x^2$$

See if you can use it here.

Oh, okay, hmm...it was on our 4-vector HW for my relativity class and with no idea how to approach it I figured it was a 4-vector problem. Thanks for the tip, I will see if I can work it out from here.

## What is the difference between u-dot and acceleration?

U-dot, also known as velocity, is a measure of an object's speed and direction. It is a vector quantity, meaning it has both magnitude and direction. Acceleration, on the other hand, measures the rate of change of an object's velocity. It is also a vector quantity, but it only measures the change in magnitude, not direction.

## How are u-dot and acceleration related?

U-dot and acceleration are related through the equation a = dv/dt, where a is acceleration, v is u-dot, and t is time. In other words, acceleration is the derivative of u-dot with respect to time.

## Which one is more important, u-dot or acceleration?

Both u-dot and acceleration are important concepts in understanding the motion of objects. U-dot tells us how fast an object is moving and in what direction, while acceleration tells us how quickly that object's u-dot is changing. They are both equally important in understanding an object's motion.

## Can u-dot be negative while acceleration is positive?

Yes, u-dot and acceleration can have different signs. In this case, the object is moving in the opposite direction of its acceleration. For example, if an object is moving forward with a positive u-dot and experiences a negative acceleration, it will slow down and eventually move in the opposite direction.

## How do u-dot and acceleration affect each other?

U-dot and acceleration are interdependent on each other. A change in u-dot will result in acceleration, and a change in acceleration will result in a change in u-dot. For example, if an object's u-dot increases, its acceleration will also increase. Similarly, if an object's acceleration decreases, its u-dot will also decrease.

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