Determining the velocity function

In summary, your original equations are incorrect. You need to express the position vector as ##\vec{r}=r\vec{i}_r(\theta)## and take into account the fact that ##\vec{i}_r## is a function of ##\theta##, that ##\theta## is a function of time, and that derivative of ##\vec{i}_r## with respect to ##\theta## can be expressed in terms of ##\vec{i}_{\theta}##.
  • #1
doktorwho
181
6

Homework Statement


Given the ## r(t) = ae^{kt}## , ##θ(t)=kt## find the velocity function that is dependent on ##r##.
##v(r)=?##

Homework Equations


3. The Attempt at a Solution [/B]
My attempt:
1)##r(t) = ae^{kt}##
2)##{\dot r(t)} = ake^{kt}##
From the first equation:
##\ln {\frac{r(t)}{a}}=\ln e^{kt}##
##\ln {\frac{r(t)}{a}}=kt##
##t=\frac{\ln {\frac{r(t)}{a}}}{k}##
Replacing the ##t## in the second equation i get:
##{\dot r}=akr##
Shouldn't this be the answer? In the answers it says ##{\dot r}=\sqrt2r##?
 
Last edited:
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  • #2
Your original equations are incorrect. ##\theta## is not a vectorl where are your unit vetors i these equations?
 
  • #3
Chestermiller said:
Your original equations are incorrect. ##\theta## is not a vectorl where are your unit vetors i these equations?
Yeah, no vectors, just the parametric equations of motion given. So what's wrong now?
 
  • #4
doktorwho said:
Yeah, no vectors, just the parametric equations of motion given. So what's wrong now?
If you are going to determine the velocity vector, you need to start out by expressing the position vector as ##\vec{r}=r\vec{i}_r(\theta)## and taking into account the fact that ##\vec{i}_r## is a function of ##\theta##, that ##\theta## is a function of time, and that derivative of ##\vec{i}_r## with respect to ##\theta## can be expressed in terms of ##\vec{i}_{\theta}##.
 
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Likes doktorwho
  • #5
Chestermiller said:
If you are going to determine the velocity vector, you need to start out by expressing the position vector as ##\vec{r}=r\vec{i}_r(\theta)## and taking into account the fact that ##\vec{i}_r## is a function of ##\theta##, that ##\theta## is a function of time, and that derivative of ##\vec{i}_r## with respect to ##\theta## can be expressed in terms of ##\vec{i}_{\theta}##.
So the polar coordinate,
##\vec r(t)=ae^{kt}\vec e_r##
##θ=kt##
##\vec v(t)=\dot r\vec e_r + r\dot θ\vec e_θ##
##\vec v(t)=ake^{kt}\vec e_r + ae^{kt}k\vec e_θ##
##v(r)=\sqrt2r##
This should be it.
 

Related to Determining the velocity function

1. What is a velocity function?

A velocity function is a mathematical representation of an object's velocity in terms of time. It describes the rate at which an object's position changes over time.

2. How do you determine the velocity function?

The velocity function is determined by taking the derivative of the position function with respect to time. This means finding the rate of change of position over time, or the slope of the position function.

3. What is the difference between velocity and speed?

Velocity is a vector quantity that includes both magnitude (speed) and direction. Speed, on the other hand, is a scalar quantity that only represents the magnitude of an object's motion.

4. What are some common units for velocity?

Velocity can be measured in many different units, but some common ones include meters per second (m/s), kilometers per hour (km/h), and miles per hour (mph).

5. How is the velocity function used in physics?

The velocity function is used in physics to analyze an object's motion and predict its future position and speed. It is also used to calculate other important quantities such as acceleration and displacement.

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