Q. No.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?
1.
The velocity and acceleration, both are parallel to \(r.\)
2.
The velocity is perpendicular to \(r\) and acceleration is directed towards the origin.
3.
The velocity is not perpendicular to \(r\) and acceleration is directed away from the origin.
4.
The velocity and acceleration, both are perpendicular to \(r.\)
Preeti reached the metro station and found that the escalator was not working. She walked up the stationary escalator in time . On other days, if she remains stationary on the moving escalator, then the escalator takes her up in time . The time taken by her to walk up on the moving escalator will be
1.
2.
3.
4.
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:
| 1. | \(5\hat{i}~\text{m/s}^2\) | 2. | \(-4\hat{i}~\text{m/s}^2\) |
| 3. | \(-8\hat{j}~\text{m/s}^2\) | 4. | \(0\) |
| 1. | The velocity and acceleration both are parallel to \(\vec{r }.\) |
| 2. | The velocity is perpendicular to \(\vec{r }\) and acceleration is directed towards to origin. |
| 3. | The velocity is parallel to \(\vec{r }\) and acceleration is directed away from the origin. |
| 4. | The velocity and acceleration both are perpendicular to \(\vec{r}.\) |
A ship \(A\) is moving westward with a speed of \(10~\text{kmph}\) 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. \(0~\text{h}\)
2. \(5~\text{h}\)
3. \(5\sqrt{2}~\text{h}\)
4. \(10\sqrt{2}~\text{h}\)
\(\vec{{R}}=4 \sin (2 \pi {t}) \hat{i}+4 \cos (2 \pi {t}) \hat{j},\)
where \(R\) is in metres, \(t\) is in seconds and \({\hat{i},\hat{j}}\) denotes unit vectors along \({x}\) and \({y}\text-\)directions, respectively. Which one of the following statements is wrong for the motion of the particle?
| 1. | Acceleration is along \((\text{-}\vec R )\). |
| 2. | Magnitude of the acceleration vector is \(\frac{v^2}{R}\), where \(v\) is the velocity of the particle. |
| 3. | Magnitude of the velocity of the particle is \(8\) m/s. |
| 4. | Path of the particle is a circle of radius \(4\) m. |
A projectile is fired from the surface of the earth with a velocity of \(5~\text{m/s}\) and at an angle \(\theta\) with the horizontal. Another projectile fired from another planet with a velocity of \(3~\text{m/s}\) 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 \(g=9.8~\text{m/s}^2\) )
1. \(3.5~\text{m/s}^2\)
2. \(5.9~\text{m/s}^2\)
3. \(16.3~\text{m/s}^2\)
4. \(110.8~\text{m/s}^2\)
The velocity of a projectile at the initial point \(A\) is \(2\hat i+3\hat j~\text{m/s}.\) Its velocity (in m/s) at the point \(B\) is:
| 1. | \(-2\hat i+3\hat j~\) | 2. | \(2\hat i-3\hat j~\) |
| 3. | \(2\hat i+3\hat j~\) | 4. | \(-2\hat i-3\hat j~\) |
1. \(\theta = \tan^{-1}\left(\frac{1}{4}\right)\)
2. \(\theta = \tan^{-1}(4)\)
3. \(\theta = \tan^{-1}(2)\)
4. \(\theta = 45^{\circ}\)
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