A moves with a speed equal to v aiming towards B. B moves vertically upwards (direction y) with a uniform speed u.
Take the frame of reference of B. B is stationary. Since A is always approaching and is directed towards B, in this frame of reference, A is traveling along a straight line (along x axis). So A does not move in y-direction. This means that, the velocity of A in direction y is also equal to u in the inertial frame of reference.
velocity of A in y direction = v_y = u
velocity of A in x direction = v_x = ?
Magnitude of speed ( velocity) of A = √(v_x² + v_y²) = v (given)
Hence v_x = √(v² - u²)
Let A meet
The distance between them is L initially.
Time taken for A to meet B = T = L / v_x = L / √(v² - u²)
Since the horizontal and vertical velocities of A are constant, A is moving in a straight line. A travels a distance of :
v * T = L * v /√(v² - u²)