Is the Curve a Line or a Point with Zero Acceleration?

In summary, the conversation discusses a curve in $\mathbb{R}^3$ with zero acceleration and shows that it must be a straight line or a point. The formula for the curve is given and it is determined that certain conditions must be met in order for the curve to be a straight line or a point.
  • #1
mathmari
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Hey! :eek:

Let $\overrightarrow{\sigma}$ a curve in $\mathbb{R}^3$ with zero acceleration. Show that $\overrightarrow{\sigma}$ is a straight line or a point.

Let $\overrightarrow{\sigma}(t)=(x(t),y(t),z(t))$ be the curve.

We have that $a(t)=\overrightarrow{\sigma}''(x''(t),y''(t),z''(t))=(0,0,0)$

So $x(t)=c_1t+c_2, y(t)=c_3t+c_4, z(t)=c_5t+c_6$

Do we have to take cases if $c_1=c_3=c_5=0$ or not to conclude that the curve is a straight line or a point?? (Wondering)
 
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  • #2
Heya! (Mmm)

mathmari said:
Do we have to take cases if $c_1=c_3=c_5=0$ or not to conclude that the curve is a straight line or a point?? (Wondering)

Yep. (Nod)
 
  • #3
I like Serena said:
Heya! (Mmm)
Yep. (Nod)
Great! Thanks! (Smile)
 

FAQ: Is the Curve a Line or a Point with Zero Acceleration?

Is a curve always a line or a point?

No, a curve can take on many different shapes and forms. It is not limited to being just a line or a point.

How do you determine if a curve is a line or a point?

This depends on the specific equation or graph of the curve. Generally, a line has a constant slope and a point has no slope. However, there are other factors to consider such as the degree of the curve and the behavior of the curve at different points.

Can a curve be both a line and a point?

No, a curve can only be one or the other. However, a curve can have a point somewhere along its length, but this does not mean the entire curve is a point.

Why is it important to understand the difference between a line and a point on a curve?

Understanding the difference between a line and a point on a curve can help in analyzing and interpreting data or equations. It can also aid in solving mathematical problems involving curves.

Are there real-life examples of curves that are neither a line nor a point?

Yes, there are many real-life examples of curves such as the shape of a river, the trajectory of a thrown ball, and the growth of a population. These curves can have complex shapes and are not limited to being just a line or a point.

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