homework and exercises - Doubt in a question related to Newton's laws of motion

In a book called Concept of Physics HC Verma Part 1 I came across this question

I can't understand why the answer is not $2u\cos θ$ (by taking of each string speed along $+y$ axis), the given answer is $u/\cos θ.$

Answer

I think you are getting confused with the balancing of forces. If mass M is not accelerating, then the forces on it are balanced. If the tension in the string is $T$ then $2T\cos\theta=Mg$. You are perhaps thinking that the velocities must be related in the same way : $2u\cos\theta=v$, the velocity of M. This is not correct. The velocities are not related to the forces - it is $F=ma$ not $F=mv$.

What is harder to see is why $u=v\cos\theta$ is correct rather than $v=u\cos\theta$.

Suppose that in a short time interval M moves up to point Y. The length of the string shortens from AM to AY. Since the angle MAY is small, AX and AY are approximately equal. Therefore the rope has shortened by distance MX in the same time that M has moved a distance MY. The speeds of M and P (the other end of the rope going over the pulley) are in the ratio $MY/MX=v/u$. From geometry $MX=MY\cos\theta$ therefore $v=u(MY/MX)=u/cos\theta$.

Alternatively you can imagine that the motion of the rope occurs in 2 stages : 1st it rotates a little about A, so that M moves to N such that AN=AM; then it moves towards A with velocity u, ending at Y. The rope has shortened by distance NY in the same time that M has moved to Y. If angle MAY is small then angles AMY and MYN are approximately equal. So $v/u=MY/NY=1/\cos\theta$ from which $v=u/\cos\theta$.