Relative Velocity and vectors

In summary, the conversation discusses the problem of two particles being shot out from a given point with equal speed in orthogonal directions in a given inertial frame. The speed of each particle relative to the other is expressed as Ur=u*[2-{(u^2)/(c^2)}]^0.5. The concept of Lorentz transformations and how velocities transform in special relativity are also mentioned.
  • #1
pinky86
4
0
I'm having trouble picturing the following problem.

In a given inertial frame, two particles are shot out simultaneously from a given point, with equal speed u, in orthogonal directions.

a) Without using 4-vectors show that the speed of each particle relative to the other is given by:

Ur=u*[2-{(u^2)/(c^2)}]^0.5
 
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  • #2
Call the given frame S'. In that frame, imagine that one particle (B) is moving with velocity +u along the y-axis; the other particle (A) is moving with velocity -u along the x-axis.

Now view things from the frame of particle A, which you can call S. Note that S' moves with speed +u along the x-axis in frame S. (The usual set up for Lorentz transformations.)

How do velocities transform in special relativity?
 
  • #3


I can understand that it can be challenging to visualize problems involving relative velocity and vectors. However, with a little bit of understanding, we can solve this problem without the use of 4-vectors.

Firstly, let's define the given inertial frame as a reference frame in which the laws of physics hold true. In this frame, two particles are shot out simultaneously from a given point, with equal speed u, in orthogonal directions. This means that the particles are moving at right angles to each other.

Now, in order to determine the speed of each particle relative to the other, we need to consider the motion of each particle separately. Let's call the particles A and B, with A moving in the x-direction and B moving in the y-direction.

Using basic vector addition, we can calculate the resultant velocity of the two particles. The magnitude of the resultant velocity, let's call it Vr, can be calculated using the Pythagorean theorem as follows:

Vr = √(VxA^2 + VxB^2)

Since the particles are moving at equal speeds u, we can substitute this value for both VxA and VxB:

Vr = √(ux^2 + uy^2)

Now, since the particles are moving at right angles to each other, we can use the trigonometric identity sin^2θ + cos^2θ = 1 to rewrite this equation as:

Vr = u√(sin^2θ + cos^2θ)

Substituting the values for sinθ and cosθ (since they are orthogonal directions) we get:

Vr = u√(1 + 1) = u√2

Therefore, the speed of each particle relative to the other is given by:

Ur = Vr - u = u√2 - u = u(√2 - 1)

We can also express this in terms of the speed of light, c, as:

Ur = u(√2 - 1) = u√(2 - (u^2/c^2))

Hence, we have shown that the speed of each particle relative to the other is given by:

Ur = u√(2 - (u^2/c^2))

I hope this explanation helps you better understand the problem and its solution. Remember, with a little bit of practice and understanding, you can easily visualize and solve problems involving
 

1. What is relative velocity?

Relative velocity is the velocity of an object in relation to another object. It takes into account the motion of both objects and measures the velocity of one object with respect to the other.

2. How is relative velocity calculated?

Relative velocity is calculated by subtracting the velocity of one object from the velocity of the other object. This can be done using vector addition or by using the relative velocity formula, which takes into account the direction and magnitude of the velocities.

3. What are vectors in relation to relative velocity?

Vectors are quantities that have both magnitude and direction. In the context of relative velocity, vectors are used to represent the velocities of the objects in motion. They are essential in calculating relative velocity and understanding the motion of objects with respect to each other.

4. Can relative velocity be negative?

Yes, relative velocity can be negative. This occurs when the motion of the objects is in opposite directions. In this case, the relative velocity will have a negative magnitude, indicating that the objects are moving away from each other.

5. How is relative velocity different from absolute velocity?

Absolute velocity is the velocity of an object with respect to a fixed reference point. It does not take into account the motion of other objects. On the other hand, relative velocity considers the motion of both objects and measures the velocity of one object with respect to the other.

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