Vector form of position, velocity, acceleration, and force

[songoku]

Homework Statement

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

a. find v in vector form

b. find mass

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

d. find v when t = 6

Homework Equations

F = ma

The Attempt at a Solution

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

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

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

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

$$|F \; \text{total}|=\sqrt{9^2+1^2}=\sqrt{82}$$

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

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

Thx

 

[rock.freak667]

yep looks correct to me.

 

[kuruman]

This problem troubles me. Newton's Second Law is a vector equation

F = ma and implies that

F_x=ma_x and F_y=ma_y

from which we get that

m = F_x/a_x = F_y/a_y

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

 

[songoku]

Thx rock.freak667

Another question : Is it possible if

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

So, the velocity :

$$v=\left(\begin{array}{cc}\frac{3}{5}\\20t\end{array}\right)$$

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)

 

[kuruman]

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

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

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.

 

[kuruman]

Addendum to my previous post

This is a bad problem.

 

[songoku]

Hi kuruman

So it should be :

$$m=\frac{F_x}{a_x}=\frac{F_y}{a_y}=\frac{|F|}{|a|}\; ?$$

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

 

[kuruman]

Hi kuruman

So it should be :

$$m=\frac{F_x}{a_x}=\frac{F_y}{a_y}=\frac{|F|}{|a|}\; ?$$

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

F_x/a_x should be equal to to F_y/a_y 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))=33.7^o

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

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.

 

[songoku]

Hi kuruman

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

Thx a lot for your help ^^