Vector form of position, velocity, acceleration, and force

In summary, the given problem presents an inconsistent scenario with the given information. The mass, acceleration, and velocity calculations based on the given force do not align with Newton's Second Law. Additionally, the problem does not mention any other forces that could account for this inconsistency. Therefore, this is a flawed problem that does not accurately represent the principles of physics.
  • #1
songoku
2,266
319

Homework Statement


[tex]x=\left(\begin{array}{cc}\frac{3}{5}t^2+2t\\10t^2+1\end{array}\right) , 0\leq t \leq 5 \; \text {and} \; F=\left(\begin{array}{cc}3\\1\end{array}\right)[/tex]

a. find v in vector form

b. find mass

c. when t = 5, there is addition of [tex]F=\left(\begin{array}{cc}t\\0\end{array}\right)[/tex]. Find the acceleration when t = 6

d. find v when t = 6

Homework Equations


F = ma

The Attempt at a Solution


a. [tex]v=\left(\begin{array}{cc}\frac{6}{5}t+2\\20t\end{array}\right)[/tex]b. [tex]|F|=\sqrt{3^2+1^2}=\sqrt{10}[/tex]

[tex]a=\left(\begin{array}{cc}\ 6/5 \\20\end{array}\right)[/tex]

[tex]|a|=\sqrt{\left(\frac{6}{5}\right)^2+20^2}\approx 20.04[/tex]

[tex]m=\frac{|F|}{|a|}=\frac{\sqrt{10}}{20.04}\approx 0.158\; kg[/tex]c. [tex]F \; \text{total}=\left(\begin{array}{cc}3\\1\end{array}\right)+\left(\begin{array}{cc}t\\0\end{array}\right) = \left(\begin{array}{cc}3\\1\end{array}\right)+\left(\begin{array}{cc}6\\0\end{array}\right)=\left(\begin{array}{cc}9\\1\end{array}\right)[/tex]

[tex]|F \; \text{total}|=\sqrt{9^2+1^2}=\sqrt{82}[/tex]

[tex]|a|=\frac{|F|}{m}\approx 57.31 \;ms^{-2}[/tex]d. [tex]v=\left(\begin{array}{cc}46/5\\120\end{array}\right)[/tex]

[tex]|v|=\sqrt{\left(\frac{46}{5}\right)^2+120^2}\approx 120.35 \;ms^{-1}[/tex]Do I get it right ?

Thx
 
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  • #2
yep looks correct to me.
 
  • #3
This problem troubles me. Newton's Second Law is a vector equation

F = ma and implies that

Fx=max and Fy=may

from which we get that

m = Fx/ax = Fy/ay

I agree that a = (6/5, 20) so if F = (3, 1), this simply doesn't work.
 
  • #4
Thx rock.freak667

Another question : Is it possible if

[tex]x=\left(\begin{array}{cc}\frac{3}{5}t\\10t^2+ 1\end{array}\right)[/tex]

So, the velocity :

[tex]v=\left(\begin{array}{cc}\frac{3}{5}\\20t\end{array}\right)[/tex]

i.e. the velocity is constant in x-direction and depends on t in y-direction?

I think it's possible such in projectile motion, where the velocity in x-direction is constant and changing in y-direction. But I'm not sure...

Thx :)

EDIT :
Sorry, I just read kuruman's post. Yes that makes sense and now I'm confused...or maybe F = ma doesn't imply that Fx=max and Fy=may ? (just guessing)
 
  • #5
You are correct. It is entirely possible to have one component depend on time and not the other. As you say, this is the case with projectile motion

[tex]
v = \left(\begin{array}{cc}v_{0x}\\v_{0y}-gt\end{array}\right).
[/tex]

F = ma is a vector equation. It says that "the vector on the left is the same as the vector on the right". When are two vectors the same? When their x components are the same and their y components are the same. Whoever authored this problem over-constrained it so that the bottom line is inconsistent with Newton's Second Law.
 
  • #6
Addendum to my previous post

This is a bad problem.
 
  • #7
Hi kuruman

So it should be :

[tex]m=\frac{F_x}{a_x}=\frac{F_y}{a_y}=\frac{|F|}{|a|}\; ?[/tex]

Another question :
Maybe it is also possible to have acceleration that is constant in x-direction and changing in y-direction? If so, the force will also constant in x-direction and changing in y-direction?

Thx
 
  • #8
songoku said:
Hi kuruman

So it should be :

[tex]m=\frac{F_x}{a_x}=\frac{F_y}{a_y}=\frac{|F|}{|a|}\; ?[/tex]

Another question :
Maybe it is also possible to have acceleration that is constant in x-direction and changing in y-direction? If so, the force will also constant in x-direction and changing in y-direction?

Thx

You are correct on both accounts. But the ratio

Fx/ax should be equal to to Fy/ay and equal to |F|/|a| at all times, no matter what the time dependence of the individual components is.

Stated differently:

The angle between the acceleration vector and the x-axis is
arctan(20/(6/5))=33.7o

The angle between the force vector and the x-axis is at all times
arctan(1/3) = 18.4o

Conclusion: The acceleration vector does not point in the same direction as the force. What does one make of this?

If the given force is the only one acting on the mass, then it is a bad problem because F is the net force and must be in the same direction as a. If F is not the net force and there is another force that tips the acceleration vector relative to the given force, the problem mentions no such force or asks you to find it. Still bad problem.
 
Last edited:
  • #9
Hi kuruman

After reading your explanation, I agree that this is bad problem

Thx a lot for your help ^^
 

1. What is the vector form of position?

The vector form of position is a mathematical representation of an object's location in space, which includes both magnitude and direction. It is typically denoted as a vector with three components: x, y, and z coordinates.

2. How is the vector form of velocity different from the vector form of position?

The vector form of velocity describes the rate of change of an object's position over time. Unlike the vector form of position, which includes both magnitude and direction, the vector form of velocity only includes direction and speed (magnitude of velocity).

3. What is the relationship between the vector forms of acceleration and velocity?

The vector form of acceleration is the rate of change of an object's velocity over time. In other words, it describes how an object's speed and/or direction is changing. This means that acceleration and velocity are closely related, as changes in velocity cause changes in acceleration.

4. How is the vector form of force related to acceleration?

The vector form of force is directly related to acceleration through Newton's Second Law of Motion, which states that the net force acting on an object is equal to its mass multiplied by its acceleration. This means that a greater force applied to an object will result in a greater acceleration.

5. Can the vector forms of position, velocity, acceleration, and force be used in any situation?

Yes, the vector forms of position, velocity, acceleration, and force can be used to describe the motion of any object in any situation, as long as the object is experiencing linear motion (motion in a straight line). These forms can also be used to analyze the motion of an object in three-dimensional space.

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