Preeti reached the metro station and found that the escalator was not working. She walked up the stationary escalator in time $t_1$. On other days, if she remains stationary on the moving escalator, then the escalator takes her up in time $t_2$. The time taken by her to walk up on the moving escalator will be 

(1)$\frac{t_1-t_2}{2}$

(2)$\frac{t_1{t}_2}{t_2-t_1}$

(3) $\frac{t_1{t}_2}{t_2+t_1}$

(4) $t_1-t_2$

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?

In the given figure, \(a=15\) m/s² represents the total acceleration of a particle moving in the clockwise direction in a circle of radius \(R=2.5\) m at a given instant of time. The speed of the particle is:

              

1. \(4.5\) m/s
2. \(5.0\) m/s
3. \(5.7\) m/s
4. \(6.2\) m/s

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 ship \(A\) is moving westward with a speed of \(10\) kmph and a ship \(B\), \(100 ~\text{km}\) South of \(A\), is moving northward with a speed of \(10\) \(\text{kmph}\). The time after which the distance between them becomes the shortest is:
1. \(0\) h
2. \(5\) h
3. \(5\sqrt{2}\) h
4. \(10\sqrt{2}\) h

The position vector of a particle \(\vec{R}\) as a function of time \(t\) is given by;
\(\overrightarrow{\mathrm{R}}=4 \sin (2 \pi \mathrm{t}) \hat{\mathrm{i}}+4 \cos (2 \pi \mathrm{t}) \hat{\mathrm{j}}\)
Where \(R\) is in meters, \(t\) is in seconds and \(\mathrm{\hat{i},\hat{j}}\) denotes unit vectors along \(\mathrm{x}\) and \(\mathrm{y}\)-directions, respectively. Which one of the following statements is wrong for the motion of the particle?

A projectile is fired from the surface of the earth with a velocity of \(5\) ms^–1 and at an angle \(\theta\) with the horizontal. Another projectile fired from another planet with a velocity of \(3\) ms^–1 at the same angle follows a trajectory that is identical to the trajectory of the projectile fired from the Earth. The value of the acceleration due to gravity on the other planet is: (given \(g=9.8\) ms^–2)
1. \(3.5\) m/s²
2. \(5.9\) m/s²
3. \(16.3\) m/s²
4. \(110.8\) m/s²

The velocity of a projectile at the initial point \(A\) is \(2\hat i+3\hat j~\)m/s. Its velocity (in m/s) at point \(B\) is:
              

The horizontal range and the maximum height of a projectile are equal. The angle of projection of the projectile is:
1. $\theta ={\tan }^{-1}\left(\frac{1}{4}\right)$
2. $\theta ={\tan }^{-1}\left(4\right)$
3. $\theta ={\tan }^{-1}\left(2\right)$
4. $\theta ={45}^0$