In a two-dimensional motion, instantaneous speed \(v_0\) is a positive constant. Then which of the following is necessarily true?

The position vector of a particle \(\overrightarrow r\) as a function of time \(t\) (in seconds) is \(\overrightarrow r=3 t \hat{i}+2t^2\hat j~\text{m}\). The initial acceleration of the particle is:
1. \(2~\text{m/s}^2\)
2. \(3~\text{m/s}^2\)
3. \(4~\text{m/s}^2\)
4. zero

The \(x\) and \(y\) coordinates of the particle at any time are \(x=5 t-2 t^2\) and \({y}=10{t}\) respectively, where \(x\) and \(y\) are in meters and \(\mathrm{t}\) in seconds. The acceleration of the particle at \(\mathrm{t}=2\) s is:

A body is moving with a velocity of \(30~\text{m/s}\) towards the east. After \(10~\text s,\) its velocity becomes \(40~\text{m/s}\) towards the north. The average acceleration of the body is:
1. \( 7~\text{m/s}^2\)
2. \( \sqrt{7}~\text{m/s}^2\)
3. \(5~\text{m/s}^2\)
4. \(1~\text{m/s}^2\)

If the position of a particle varies according to the equations \(x= 3t^2\)${\mathrm{}}^{}$, \(y =2t\), and \(z= 4t+4\), then which of the following is incorrect?

A particle is moving in the \(XY\) plane such that \(x = \left(t^2 -2t\right)~\text m,\) and \(y = \left(2t^2-t\right)~\text m,\) then:

A particle moves so that its position vector is given by \(r=\cos \omega t \hat{x}+\sin \omega t \hat{y}\) where \(\omega\) is a constant. Based on the information given, which of the following is true?

A particle moves in space such that:
\(x=2t^3+3t+4;~y=t^2+4t-1;~z=2\sin\pi t\)
where \(x,~y,~z\)  are measured in meters and \(t\) in seconds. The acceleration of the particle at \(t=3\) seconds will be:

A particle is moving along a curve. Select the correct statement.

A particle moving on a curved path possesses a velocity of \(3\) m/s towards the north at an instant. After \(10\) s, it is moving with speed \(4\) m/s towards the west. The average acceleration of the particle is: