Homework Hint: Problem 2/239

Note that the "Given" information on the motion of the rocket is in terms of Cartesian components:  v = vj and a = aj. Note that v is unknown and a is known.

The "Find" information of the motion is in terms of polar components. Your task is to transform one set of components to the other set. There are a number of ways to do this. One way is to project the velocity and acceleration vectors onto the e_r and e_theta directions and solve:

• Write e_r and e_theta in terms of i and j:  e_r = cos(theta) i + sin(theta) j and e_theta = -sin(theta) i + cos(theta) j
• Project v onto e_r (using v • e_r). Here "•" is the dot product, not a multiplication.
• Project v onto e_theta (using v • e_theta).
• Solve the above two equations for v and r_dot using the known value for theta_dot.
• Project a onto e_r and solve for r_dot_dot.
• Project a onto e_theta and solve for theta_dot_dot.

Another way would be to write j in terms of e_r and e_theta. Substitute j into the above given expressions for v and a. Set these expressions equal to the polar expressions for v and a. Balance coefficients and solve. 

And there are other ways... Choose the method that works best for you.