Trying to calculate relative speed between two moving points

In summary, two points on a 2D rectangular coordinate system, P and M, have positions that are functions of time. The distance between them, R(t), can be calculated using the formula R(t) = √((p_x - m_x)^2 + (p_y - m_y)^2). The relative speed between the points, V_{pm}(t), can be found by taking the derivative of R(t) with respect to time. However, explicit expressions for the components of P(t) and M(t) are needed to calculate the derivative. The minus sign in the formula for relative speed is based on intuition and may not always be accurate. Vector addition can be used to find the relative velocity between two points,
  • #1
hkBattousai
64
0
There are two points on the 2D rectangular coordinate system, namely P and M.

Their positions are function of time and are:
[tex]Position \, of \, P: \, (p_x(t), \, p_y(t))[/tex]
[tex]Position \, of \, M: \, (m_x(t), \, m_y(t))[/tex]

Distance between them is:
[tex]R(t) \, = \, \sqrt{(p_x - m_x)^2 \, + \, (p_y - m_y)^2}[/tex]

And the relative speed (magnitude of relative velocity) between them is:
[tex]V_{pm}(t) \, = \, - \frac{dR}{dt}[/tex]

Is it correct up to this step?

If so, can you please help me take this derivative?
If not, how do I calculate this relative speed?

Any help will be appreciated.
 
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  • #2
To get the derivative you need explicit expressions for the components of P(t) and M(t). Based on the information presented, all you can get is an expression involving the derivatives of the components.

Aside: why the minus sign for the relative speed?
 
  • #3
mathman said:
To get the derivative you need explicit expressions for the components of P(t) and M(t). Based on the information presented, all you can get is an expression involving the derivatives of the components.
There are no explicit expression. Speeds of these points are arbitrary.

mathman said:
Aside: why the minus sign for the relative speed?
Man's intuition :)
Think of it a little, correct me if I'm wrong...
 
  • #4
Try something like y1= x^2 and y2 = x

let the x component velocities be the same for both functions then the y component velocities of the functions at any point as respects x would be the derivative of the functions at those points (slope) and the relative velocities of those point are the difference of the y component velocites.

I don't think the distance between the 2 points has anything to do with relative velocities since the 2 function above cross at x = 1.

I think that is correct but correct me if I am wrong.
 
  • #5
jim pohl
dont see how vector addition of derivatives can find relative velocites between points could you give more details?

My understanding is that vector addiiton gives a new vector but then that new vector only represents a direction and magnitude but as I already pointed out the distance beween two points has nothing to do with relative velocity of those points - maybe I am not understanding your post i will study further.

later: do you mean the vector addition of the y and x component velocites? OK maybe thatll work - need to go over this
 
Last edited:
  • #6
scan0002.jpg


(If you have trouble reading this try holding down CTRL key while you move the scroll wheel on your mouse.)

A particle travels along vector A in the direction shown and crosses vector C at point a
Another particle travels along vector B in the direction shown and crosses vector C at point b
What is the relative velocity of points a, b if the particle along vector A has a velocity
of 2 feet per second and the particle along vector B has a velocity of 3 feet per second?

(using degrees in this example)

V = VA cos 110 + VB cos 40 = 2 cos 110 + 3 cos 40 = 1.61409 feet per second

The relative velocity of points a,b can also be zero. For the above triangle, any positive velocity along vector A will have a velocity along vector B such that the relative velocity of a,b will equal zero.

Example

The particle along vector A has a velocity of 2 feet per second as it did before. What is the velocity of the particle along vector B so relative velocity of points a, b equal zero?

-2 cos 110 / cos 40 = .89295 feet per second.

so relative velocity of 2 moving points are dependant on the velocity of a particle along the vector and the angles between vectors.

Possible implications.

An observer at point b looks along vector C and notices that a certain star at point a does not have a red shift. But that star could be traveling at .2C along vector A while the observer is traveling at .089295C along vector B. Einstein did indicate that there was no absolute motion. The reasoning is that the relative velocity of objects are dependant on their vector directions as well as the velocity along the vectors.
 

1. What is relative speed?

Relative speed is the speed at which an object or person is moving relative to another object or point of reference. It takes into account the movement of both objects and is often measured in terms of distance per unit time, such as miles per hour or meters per second.

2. How do you calculate relative speed?

To calculate relative speed between two moving points, you need to first determine the speed of each point relative to a fixed point or reference frame. Then, you can subtract the speeds to find the relative speed between the two points. This can be done using simple algebraic equations or with the help of vector calculations.

3. Can relative speed be negative?

Yes, relative speed can be negative. This indicates that the two objects are moving in opposite directions or that one is moving faster than the other. It is important to pay attention to the sign of relative speed as it can affect the overall calculation and interpretation of the data.

4. What factors can affect relative speed calculations?

The accuracy of relative speed calculations can be affected by a variety of factors, such as the precision of the measurements, the accuracy of the timing, and the choice of reference frame. Other factors, such as wind resistance or friction, can also impact the actual speed of an object and therefore affect the relative speed calculation.

5. Why is calculating relative speed important?

Calculating relative speed is important in many fields of science and engineering, such as physics, astronomy, and transportation. It allows us to understand the relationships between moving objects and make predictions about their future movements. It also helps us to design and optimize systems for efficient and safe movement, such as in the development of vehicles and spacecraft.

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