Finding speed of mass m attached with two inextensible strings [duplicate]

Question

Question:In the arrangement shown in figure,the ends P and Q of an inextensible string move downwards with uniform speed u. Pulleys A and B are fixed. The mass M moves upwards with a speed-

Actual solution: Let the speed of mass M be v.The length of the string will remain constant. So by constraint relationship, we can say that the velocities increasing the length of the string (u)= velocities decreasing the length of string.( Component of v on any one of left or right string= v cosθ)

Therefore, v cosθ= u and hence v= u/cosθ

My approach:: I did just the reverse of this. I took components of u along the y-axis to get the speed of mass m. Here is the reason why:

The velocity of the mass M should be the vector-summation of the velocities of the string attached to block A and the string attached to block B.After all, these strings are the cause behind the motion of the mass M and thus, their "influence" must be added in order to obtain the behavior of mass M. As the motion of mass m is vertically upwards, their components must cancel along the X-axis and should add up on the Y-axis. Component of left string on Y-axis= u cosθ and component on Y-axis=u cosθ. So net resultant and the speed of mass m=2u cosθ.

Where am I wrong as the answer is not 2u cosθ?

Answer 1

The question states that both strings are moving at $u$. Note the symmetry in the problem, if both strings were moving at different rates your mass would drift to one side. So you can think of it as the mass M being pulled up evenly at the same rate from both sides. You may be confusing the velocities with a force. For example, if we are talking about the mass M when it is at rest and we start pulling then we add the forces from both strings because they are contributing to the acceleration of the mass. But in this case there is no acceleration, we are told that the strings are already moving at a constant velocity. Therefore by constraint, the mass M which is connected to the strings must be moving a the same rate as the strings. If it were moving a twice the speed then your mass might be moving faster than the strings for some angle which is not right.