Vector Calculus Answers

For the velocity potential function, φ = x^2 - y^2 . Find the velocity components at the point (4, 5).

If for a two dimensional potential flow, the velocity potential is given by : φ = 4x(3y - 4), determine the velocity at the point (2, 3). Determine also the value of stream function ψ at the point (2, 3).

If F=A×(B×C) where A=5it^2 + (3t-2)j + 6tk; B=2i+4tj+3(1-t)k and C=4ti+5jt^2 -3tk evaluate integral of Fdt with limits zero to one.

What do you understand by unit tangent vector

The position vector r of a particle at time t is
r = 2t
2
i + (t
2 − 4t)j + (3t − 5)k.
Find the velocity and the acceleration of the particle at time t. Show that when t =
2
5
the velocity
and the acceleration are perpendicular to each other. The velocity and the acceleration are resolved
into components along and perpendicular to the vector i − 3j + 2k. Find the velocity and acceleration
components parallel to this vector when t =
2
5

1. The position vector r of a missile at any time t is given by
r = ti + (8t − t
2
)j,
where i and j are unit vectors in the horizontal and upward vertical directions. Find its
(a) average velocity from t = 2 to t = 3,
(b) velocity, speed and direction of motion when t = 4,
(c) position vector when it is moving parallel to the vector i − 2j,
(d) acceleration vector.
2. The position vector r of a moving particle at time t after the start of the motion is given by
r = 5(1 + 4t)i + 5(19 + 2t − t
2
)j.
Find the initital velocity of the particle. At time t = T, the particle is moving at right angles to its initial
direction of motion. Find the value of T and the distance of the particle from its initial position at this
time.