Problem 1:
The position vector of a point is given by,
$\vec{r}=(\frac{4}{3}t^3-2t)\hat{i}+t^2\hat{j}$
Find the velocity and acceleration of the point at t=3sec. [The distance is measured in meters]

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Problem 2:
The motion of a particle is described by the equation, $x=4\sin{2t}$, $y=4\cos{2t}$ and $z=6t$. Find the equation of velocity and acceleration of the particle.

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Problem 3:
A particle moves along a curve $x=2\sin{3t}$, $y=2\cos{3t}$ and $z=8t$. At any instant of time (t>0), find the velocity and acceleration of the particle.

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Problem 4:
The velocity of a moving particle at any instant of time is given by,
$\displaystyle{\vec{v}=2\hat{i}+5t\hat{j}+\frac{1}{t}\hat{k}}$
Find the position vector of that particle at that instant of time.

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Problem 5:
A particle is moving along a curve in a plane. Derive an expression for the displacement, radial and transverse components of velocity and acceleration. Prove that for the motion of a particle in a plane, $\vec{v}=\dot{r}\hat{r}+r\dot{\theta}\hat{\theta}$ and
$\vec{a}=(\ddot{r}-r\ {\dot{\theta}}^2)\hat{r}+(\ddot{r}\theta+2r\ddot{\theta})\hat{\theta}$

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Problem 6:
The trajectory of a particle moving in a plane in a straight line passing through the origin. What is the transverse component of velocity?

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Problem 7:
A particle moves in a plane in such a way that its distance from the origin remains constant. What is the radial component of the velocity?

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Problem 8:
The path of a projectile is defined by the equation $\displaystyle{r=3t-\frac{1}{30}t^2}$ and ${\theta}^2=1600-t^2$. Find its velocity and acceleration after 30 sec.

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Problem 9:
For planar motion, $x=r\ \cos\theta$ and $y=r\ \sin\theta$. Prove that
$\displaystyle{\dot{r}=\frac{x\dot{x}+y\dot{y}}{r}}$
and, $\displaystyle{r\dot{\theta}=\frac{x\dot{y}-y\dot{x}}{r}}$

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Problem 10:
A point moving in a plane has co-ordinates x=3 and y=4.  And has components of speed $\dot{x}=5$ m/sec and $\dot{y}=8$ m/sec at some instant of time. Find the components of speed in polar co-ordinates $r$, $\theta$ along the direction $\hat{r}$ and $\hat{\theta}$ respectively.

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Problem 11:
The polar co-ordinates of a point $(r,\ \theta,\ \phi)$ = $(8,\ 30^{\circ},\ 45^{\circ})$. Find the Cartesian co-ordinates of the same point.

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Problem 12:
The spherical polar co-ordinates of a point are $(16,\ 60^{\circ},\ 30^{\circ})$. Find the Cartesian co-ordinate of that point.

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Problem 13:
The Cartesian co-ordinates of a point are $(1,\ 0,\ 0)$. Find the spherical polar co-ordinates of that point.

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Problem 14:
The Cartesian co-ordinates of a point are $(1,\ 0,\ 1)$. Find the spherical polar co-ordinates of that point.

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Problem 15:
Calculate the spherical polar co-ordinates of a point whose Cartesian co-ordinates are $(1,\ 0\ \sqrt{3})$.

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Problem 16:

The polar co-ordinates of a particle moving in a plane are given by, $r=a\sin{\omega}_1{t}$ and $\theta={\omega}_2{t}$. Obtain an expression for the polar components of the velocity and acceleration of the particle.

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Problem 17:

Show that the solid angle subtended by a ring element cut from a sphere of radius R is given by, $d\omega=4\pi\ \sin\frac{\theta}{2}\ \cos\frac{\theta}{2}\ d\theta$, where $\theta$ is the angle between the normal through the centre of the ring and the line joining centre of the sphere with a point on the internal circumference and $d\theta$ is the angular width of the element.

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