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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