Question

Consider a meter stick at rest in a reference frame \(F\). It lies in the \(x, y\) -plane and makes an angle of \(37^{\circ}\) with the \(x\) -axis. The reference frame \(F\) then moves with a constant velocity \(v\) parallel to the \(x\) -axis of another reference frame \(F\). a) What is the velocity of the meter stick measured in \(F\) at an angle \(45^{\circ}\) to the \(x\) -axis? b) What is the length of the meter stick in \(F^{\prime}\) under these conditions?

Short answer

Answer: The velocity of the meter stick in reference frame F' at an angle of \(45^\circ\) to the \(x\)-axis is \(v\), and its length is \(1\, \text{m} \sqrt{1-\frac{v^2}{c^2}}\).

Step-by-step solution

Calculate components of velocity in \(F'\)

Since the meter stick is at rest in reference frame \(F\), its velocity in \(F'\) will have only an \(x\)-component equal to \(v\). Therefore, we have: \(v_{x'} = v\) \(v_{y'} = 0\) Step 2: Calculate the velocity of the meter stick at a \(45^\circ\) angle

Calculate velocity at \(45^\circ\) angle

We are to find the velocity of the meter stick in the reference frame \(F'\) at an angle of \(45^\circ\). We can use the following equations to determine the \(x\) and \(y\) components of the velocity at this angle: \(v_{x'45} = v_{x'} \cos{45^\circ}\) \(v_{y'45} = v_{x'} \sin{45^\circ}\) Plugging in the values from Step 1, we have: \(v_{x'45} = v \cos{45^\circ}\) \(v_{y'45} = v \sin{45^\circ}\) Step 3: Calculate the magnitude of the velocity at a \(45^\circ\) angle

Calculate the magnitude of the velocity

To find the magnitude of the velocity, we can use the Pythagorean theorem: \(v_{45} = \sqrt{v_{x'45}^2 + v_{y'45}^2}\) Plugging in the values from Step 2, we have: \(v_{45} = \sqrt{(v \cos{45^\circ})^2 + (v \sin{45^\circ})^2}\) Step 4: Simplify the expression for the magnitude of the velocity

Simplify the velocity expression

By simplifying the expression, we can find the velocity of the meter stick at a \(45^\circ\) angle: \(v_{45} = v \sqrt{\cos^2{45^\circ} + \sin^2{45^\circ}}\) Since \(\cos^2{\theta} + \sin^2{\theta} = 1\), we have: \(v_{45} = v \sqrt{1} = v\) Thus, the velocity of the meter stick in reference frame \(F'\) at an angle of \(45^\circ\) to the \(x\)-axis is equal to \(v\). Step 5: Find the length of the meter stick in reference frame \(F'\)

Find the length of the meter stick in \(F'\)

Since there is length contraction in the direction of motion, the length of the meter stick in reference frame \(F'\) will be given by the Lorentz transformation formula: \(L' = L \sqrt{1 - \frac{v^2}{c^2}}\) Here, \(L\) is the length of the meter stick in reference frame \(F\), which is \(1 \,\text{meter}\), and \(c\) is the speed of light. So, the length of the meter stick in reference frame \(F'\) is given by: \(L' = 1\, \text{m} \sqrt{1-\frac{v^2}{c^2}}\) Therefore, we have the velocity of the meter stick in reference frame \(F'\) at an angle \(45^\circ\) to the \(x\)-axis as \(v\) and its length as \(1\, \text{m} \sqrt{1-\frac{v^2}{c^2}}\).