Question

A rocket cruises past a laboratory at 1.00×106 m/s in the positive x-direction just as a proton is launched with velocity (in the laboratory frame) v⃗=(1.41×106 i^+1.41×106 j^)\vec{v} = (1.41 	imes $$10^6$$\,\hat{i} + 1.41 	imes $$10^6$$\,\hat{j}) m/s. What are the proton's speed and its angle from the y-axis in the rocket frame?

Accepted Answer

Step 1: Recognize that this is a problem involving relative velocity in special relativity. The velocities given are in the laboratory frame, and we need to transform them into the rocket frame using the relativistic velocity transformation equations. Step 2: Write down the relativistic velocity transformation equations for the x and y components of the proton's velocity. For the x-component: $$ v'_x = \frac{v_x - u}{1 - \frac{v_x u}{c^2}} $$, where $$ v_x  is $$the x-component of the proton's velocity in the lab frame, $$ u  is $$the velocity of the rocket relative to the lab, and $$ c  is $$the speed of light. For the y-component: $$ v'_y = \frac{v_y}{\gamma (1 - \frac{v_x u}{c^2})} $$, where $$ \gamma = \frac{1}{\sqrt{1 - \frac{u^2}{c^2}}} . $$Step 3: Substitute the given values into the equations. The x-component of the proton's velocity in the lab frame is $$ v_x = 1.41 	imes 10^6 \, 	ext{m/s} $$, the y-component is $$ v_y = 1.41 	imes 10^6 \, 	ext{m/s} $$, and the rocket's velocity is $$ u = 1.00 	imes 10^6 \, 	ext{m/s} . $$Use $$ c = 3.00 	imes 10^8 \, 	ext{m/s} $$ for the speed of light. Step 4: Calculate the transformed x and y components of the proton's velocity in the rocket frame using the equations from Step 2. First, compute $$ \gamma $$ using $$ \gamma = \frac{1}{\sqrt{1 - \frac{u^2}{c^2}}} . $$Then, substitute $$ \gamma $$, $$ v_x $$, $$ v_y $$, and $$ u $$ into the equations for $$ v'_x $$ and $$ v'_y . $$Step 5: Determine the proton's speed and angle in the rocket frame. The speed is given by $$ v' = \sqrt{v'_x^2 + v'_y^2} $$, and the angle from the y-axis is $$ 	heta = \arctan\left(\frac{v'_x}{v'_y}ight) . $$Use the transformed components $$ v'_x $$ and $$ v'_y $$ from Step 4 to compute these quantities.

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Key Concepts

Relative Velocity

Vector Addition

Speed and Direction