A railroad gun of mass M = 2.0 kg fires a shell of mass m =1.0 kg at an angle of theta= 35.0 with respect to the horizontal as measured relative to the gun. After the firing is complete, the final speed of the projectile relative to the gun (muzzle velocity) is v_0 =100.0 m/s . The gun recoils with speed v_r and the instant the projectile leaves the gun, it makes an angle phi with respect to the ground.

a)What is v_p , the speed of the projectile with respect to the ground?
b)What is phi , the angle that the projectile makes with the horizontal with respect to the ground (in degrees)?

To solve this problem, we can use conservation of momentum and conservation of energy principles. Let's start by solving part (a) for the speed of the projectile with respect to the ground (v_p).

Step 1: Find the velocity of the gun after firing (v_g).

Using the conservation of momentum principle, we know that the total momentum before firing is equal to the total momentum after firing.

Initial momentum of the system = (mass of the gun x initial velocity of the gun) + (mass of the shell x initial velocity of the shell relative to the gun)
Initial momentum of the system = (M x 0) + (m x v_0) [since the gun is initially at rest]

Final momentum of the system = (mass of the gun x final velocity of the gun) + (mass of the shell x final velocity of the shell relative to the gun)
Final momentum of the system = (M x -v_r) + (m x v_p) [negative sign for the velocity of the gun since it's in the opposite direction]

Therefore, we can write the equation:

m x v_0 = M x (-v_r) + m x v_p

Step 2: Find the velocity of the shell with respect to the ground (v_s) using the relative motion principle.

v_s = v_g + v_p

Step 3: Solve for v_p.

From step 1, we can rewrite the equation as:

m x v_0 - M x (-v_r) = m x v_p

Simplifying further:

v_p = (m x v_0 + M x v_r) / m

Substituting the given values, we can calculate the result.

For part (b), we need to find the angle phi that the projectile makes with the horizontal with respect to the ground.

Step 4: Find the horizontal and vertical components of the final velocity of the projectile (v_px and v_py).

v_px = v_p x cos(theta)
v_py = v_p x sin(theta)

Step 5: Find the angle phi.

phi = arctan(v_py / v_px)

Substituting the values of v_px and v_py, we can calculate the result.

Note: Remember to convert the angle from radians to degrees if required.

I hope this helps you solve the problem! Let me know if you have any further questions.