Because P is constrained along the ground, and B is going to be moving straight up, and the sling is of length L is both cases, You can find the IC of the sling itself connecting B and P using the geometry at the two positions. You can force the angular velocity of the sling to agree with B’s velocity, then solving for P. You need to do some trig in both locations but it works out.
Since the projectile has to slide along the ground before it takes off, how does that tether dragging behind the beam change how fast it’s actually moving?
If we have the velocity of B, is the following equation the correct way of solving for P? I have an I and J component for velocity for P, which doesn’t seem correct.
Yes, your equation is fine, provided that you use omega_BP (an unknown). Your vector equation has two scalar components in terms of the unknowns of v_P (where the vector for v_P is along the ground) and omega_BP.
How do we relate the speed of P in terms of W so that we can solve for w using the energy equation and then solve for vp
Because P is constrained along the ground, and B is going to be moving straight up, and the sling is of length L is both cases, You can find the IC of the sling itself connecting B and P using the geometry at the two positions. You can force the angular velocity of the sling to agree with B’s velocity, then solving for P. You need to do some trig in both locations but it works out.
Since the projectile has to slide along the ground before it takes off, how does that tether dragging behind the beam change how fast it’s actually moving?
The rope behaves as a rigid link. Consider using the usual rigid body kinematics equation for velocity.
If we have the velocity of B, is the following equation the correct way of solving for P? I have an I and J component for velocity for P, which doesn’t seem correct.
V_P = V_B + omega x r_P/B
Yes, your equation is fine, provided that you use omega_BP (an unknown). Your vector equation has two scalar components in terms of the unknowns of v_P (where the vector for v_P is along the ground) and omega_BP.