Deriving Vector and position vectors from Force vector

In summary, the problem involves calculating the position and velocity vectors of an object at rest at the origin, when a new force is applied at t=0. The force has components Fx(t) = k1 + k2y and Fy(t) = k3t, where k1, k2, and k3 are constants. The position and velocity vectors can be determined by dividing the force vector by the mass, and then integrating the resulting acceleration and velocity vectors with respect to time. However, it is important to remember to include all necessary constants when integrating.
  • #1
HclGuy
13
0

Homework Statement


An object of mass m is at rest at equilibirum at the origin. At t=0, a new force [tex]\vec{F}(t)[/tex] is applied that has components
[tex]F_{x}[/tex](t) = k1+k2y [tex]F_{y}[/tex](t)=k3t
where k1, k2, and k3 are constants. Calculate the position r(t) and velocity v(t) vectors as functions of time.

Homework Equations


We know that Force = ma.
and that :
[tex]\int \vec{a}dt = \vec{v}(t)[/tex]
[tex]\int \vec{v}dt = \vec{r}(t)[/tex]

The Attempt at a Solution



I'm not sure if I'm doing this right but
I did
[tex]\vec{F}(t)[/tex] =[tex](k1+k2y)\hat{i}[/tex]+[tex](k3t)\hat{j}[/tex]
I divided the Force vector by the scalar value of m, the mass to get [tex]\vec{a}[/tex]
[tex]\vec{a}(t)[/tex] = [tex](k1+k2y)/m\hat{i}[/tex]+[tex](k3t)/m\hat{j}[/tex]

[tex]\vec{v}(t)[/tex]=[tex]\int \vec{a}dt[/tex] = [tex](k1+k2y)t/m \hat{i}[/tex] + [tex](k3t^2)/2m \hat{j}[/tex]

then integrate the velocity vector to get the position vector, am I doing this right at all?
 
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  • #2
You reasoning seems good, but you are forgetting some constants when you integrate.
 
  • #3
Thanks, just noticed that myself as well.
 
  • #4
note that in Fx(t) = k1 + k2y, the y is not a constant, so the integral of k2y dt is not equal to k2yt
 

1. How do you determine the magnitude and direction of a force vector?

The magnitude of a force vector can be determined by using a scale or ruler to measure the length of the vector. The direction can be determined by using a protractor to measure the angle between the vector and a reference point, such as the x-axis.

2. What is a position vector and how is it related to a force vector?

A position vector is a mathematical representation of a point in space, typically represented by an arrow from the origin to the point. It is related to a force vector by the fact that the force vector acts on an object at a specific position in space, and can be represented by a position vector from the origin to that point.

3. How do you derive a force vector from a given position vector?

To derive a force vector from a given position vector, you first need to determine the magnitude and direction of the force. This can be done by using mathematical equations, such as Newton's laws of motion, which relate force to mass and acceleration. Once the magnitude and direction are known, the force vector can be represented by a position vector in the same direction and with the same length.

4. Can a force vector have a negative magnitude or direction?

Yes, a force vector can have a negative magnitude and/or direction. This indicates that the force is acting in the opposite direction of the positive vector. For example, if a force is acting in the negative x-direction, the magnitude of the force vector would be negative and the direction would be towards the negative x-axis.

5. How can vector and position vectors be used in real-world applications?

Vector and position vectors are used in many real-world applications, particularly in physics and engineering. They can be used to calculate and represent forces acting on objects, such as in the design of structures or vehicles. They are also important in navigation and mapping, as they can be used to determine the position and movement of objects in space.

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