4 from the hw 2. I added the info of missing parts

this is the question for number 4 hw2

4. Relative to Alice, a proton travels to the right with a speed of 0.8c. a) Relative to the proton, an electron travels to the left with a speed of 0.35c. What is the speed and direction of the electron relative to Alice? b) Instead, in the inertial frame of the proton, suppose the electron travels up the page in the y-direction with a speed of 0.25c. In this case, relative to Alice what is the electron's speed and what angle does the electron's velocity form with the horizontal?

This is the answer for the 4 from hw2

4. (a) To determine the speed and direction of the electron relative to Alice, we can use the relativistic velocity addition formula.

Let's denote:

v₁ = the velocity of the proton relative to Alice (0.8c)

v₂ = the velocity of the electron relative to the proton (-0.35c)

The formula for adding velocities in special relativity is:

vrel=(v1+v2)/(1+v1v2/c2)

= (0.8c - 0.35c)/(1 - 0.28c²/c²)

= 0.625c

hereforevrel=0.625c

Therefore, the speed of the electron relative to Alice is approximately 0.625 times the speed of light (c), and the direction is opposite to the direction of the proton.

(b) To determine the electron's speed and the angle it forms with the horizontal, we can use the concept of relative velocity in special relativity.

Let's denote:

v_e-p = the velocity of the electron relative to the proton (0.25c)

v_p-A = the velocity of the proton relative to Alice (0.8c)

To find the electron's velocity relative to Alice, we can use the relativistic velocity addition formula:

vrel=(vep+vpA)/(1+(vepvpA)/c2)

Substituting the given values:

v_rel = (0.25c + 0.8c) / (1 + 0.2c²/c²)

= 0.875c

hereforevrel=0.875c

Therefore, the speed of the electron relative to Alice is approximately 0.875 times the speed of light c.

To find the angle the electron's velocity forms with the horizontal, we can use trigonometry. Since the electron's velocity is in the y-direction, and we know its speed relative to Alice, we can find the angle using the tangent function:

tanheta=(vey/vex)

Since the electron's velocity is purely in the y-direction and there is no x-component, the angle will be 90 degrees (or /2 radians).

Therefore, the electron's velocity forms a right angle (90 degrees or /2 radians) with the horizontal.