Velocity of a particle GIVEN position vector

In summary, the article discusses a conversation concerning a particle's speed at time t=1. The author does not know how to solve for the particle's velocity at this time, so they ask for help from a random variable. The random variable helps them differentiate the position vector and find the speed at time t=1.
  • #1
vorcil
398
0
A [article follows the path given by the position vector
r= (4t , 3 , t^3)

what's its' speed at t=1?

no idea how to solve this
i know velocity is the derivative of the position with respect to time

do i just solve, 4, 0, 3t^2, then stick it in?
4 + 3 = 7m/s??
 
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  • #2
Velocity is a vector quantity. The particle's velocity at time t is v = (4,0,3t^2). At time t=1, the velocity of the particle is v = (4,0,3). It's speed at time t=1 is [tex]\sqrt{4^{2}+0^{2}+3^{2}}[/tex] = 5.
 
  • #3
Random Variable said:
Velocity is a vector. The particles vecolocty at time t is v = (4,0,3t^2). At time t=1, the veclocity of the particle is v = (4,0,3). It's speed at time t=1 is [tex]\sqrt{4^{2}+0^{2}+3^{2}}[/tex] = 5.


oh so it's just the magnitude of the differentiated vector?

this was the original equation
r= (4t , 3 , t^3)

so i'd differnetiate it,

(4 , 0 , 3t^2)
and find the |r| or the magnitude of it?
 
  • #4
EDIT: You want the magnitude of v, not r.
 
  • #5
Random Variable said:
Yes. Speed is a scalar.

I find it strange to differentiate the position vector

what happened to the square on the t^3, 3t^2, it became 4,0,3?
what about the t^2? or is the ^2 part of the t, and we forget the t's?
 
  • #6
You wanted the speed at time t=1.
 
  • #7
You're given the particle's postion as a function of time. If you differentiate, then you know the rate and direction at which the particle's position is changing.
 
  • #8
Random Variable said:
You wanted the speed at time t=1.

so how do i make r= (4t , 3 , t^3) become 4,0,3?
4*1t , 3*0t, 1*1^3?

i thought differentiating r= (4t , 3 , t^3) 4t -> 4 ,3 -> 0, t^3 -> 3t^2
then substituting in 1 for t, you get the velocity vector, 4,0,3?
then i'd just get |(4,0,3)|
 
  • #9
v = dr/dt = (4, 0, 3t2)

speed = |v| = [itex]\sqrt{4^2 + 0^2 + 9t^4 }[/itex]
The speed at t = 1 is the value of the radical above at t=1.
 
  • #10
vorcil said:
so how do i make r= (4t , 3 , t^3) become 4,0,3?
4*1t , 3*0t, 1*1^3?

i thought differentiating r= (4t , 3 , t^3) 4t -> 4 ,3 -> 0, t^3 -> 3t^2
then substituting in 1 for t, you get the velocity vector, 4,0,3?
then i'd just get |(4,0,3)|

(4,0,3) is a vector with magnitude 5.
 
  • #11
Random Variable said:
(4,0,3) is a vector with magnitude 5.
yeah thanks i know that, but how did you get (4,0,3) from r= (4t , 3 , t^3)?
 
  • #12
Random Variable said:
Velocity is a vector quantity. The particle's velocity at time t is v = (4,0,3t^2). At time t=1, the velocity of the particle is v = (4,0,3). It's speed at time t=1 is [tex]\sqrt{4^{2}+0^{2}+3^{2}}[/tex] = 5.


OH RIGHT,
thanks Random variable, XD XD XD :rofl:

Just to confirm, i differentiate r= (4t , 3 , t^3)?
 
  • #13
vorcil said:
OH RIGHT,
thanks Random variable, XD XD XD :rofl:

Just to confirm, i differentiate r= (4t , 3 , t^3)?

Hasn't this been answered in post 9?
 

What is the definition of velocity?

Velocity is a measure of how fast an object is moving and in what direction. It is a vector quantity, meaning it has both magnitude (speed) and direction.

How is velocity calculated?

Velocity is calculated by dividing the change in position (or displacement) by the change in time. This can be represented mathematically as v = Δx/Δt, where v is velocity, Δx is change in position, and Δt is change in time.

What is the difference between velocity and speed?

The main difference between velocity and speed is that velocity is a vector quantity, while speed is a scalar quantity. This means that velocity includes direction, while speed does not.

Can velocity be negative?

Yes, velocity can be negative. This indicates that the object is moving in the opposite direction of the chosen reference point. For example, if a car is moving east and then turns around and starts moving west, its velocity will be negative.

How is velocity represented graphically?

Velocity can be represented graphically on a position-time graph, where the slope of the line represents the velocity. A steeper slope indicates a greater velocity, while a flatter slope indicates a slower velocity.

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