Q.261. Velocity of a particle constrained to move as per conditions given.

Question 261.
Two particles A and B are confined to always be in a circular groove of radius √(17) m as shown in the following figure. At the same time, the particles must also be in a slot that has the shape of a parabola, which has an equation x = y^2 at t = 0. The slot is shown dashed at time t = 0. If the slot moves to the right at a constant speed of 3 m/s, find the speed of A at t = 1 second.

Answer 261.
At time t = 1, x = 3 m
=> x - 3 = y^2 ... (because the particle A is in the parabolic slot)
Also, x^2 + y^2 = 17
Solving these two equations, x = 4 and y = 1.

As the particle is always on the circle
x^2 + y^2 = 17
=> 2x dx/dt + 2y dy/dt = 0
=> x dx/dt + y dy/dt = 0
=> 4u + v = 0 ... (1)
[taking dx/dt = u and dy/dt = v, x = 4 and y = 1]

Also, as the particle A is in the parabolic slot at any time t,
x - 3t = y^2
=> dx/dt - 3 = 2y dy/dt
=> u - 3 = 2v
=> u - 2v = 3 ... (2)

Solving eqns. (1) and (2),
u = 1/3 and v = - 4/3
=> velocity of the particle
= √(u^2 + v^2)
= √[(1/3)^2 + (-4/3)^2]
= √(17) / 3 m/s.

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