Q. No.Let \(\mathrm{ABCDEF}\) be a regular hexagon, with the vertices taken in order. The resultant of the vectors: \(\overrightarrow{AB},~\overrightarrow{BC},~\overrightarrow{CD},~\overrightarrow{DE}\) equals, in magnitude, the vector:
1. \(\overrightarrow{AB}\)
2. \(\overrightarrow{AD}\)
3. \(\sqrt2\overrightarrow{AB}\)
4. \(\sqrt3\overrightarrow{AB}\)
1. \(\overrightarrow{AB}\)
2. \(\overrightarrow{AD}\)
3. \(\sqrt2\overrightarrow{AB}\)
4. \(\sqrt3\overrightarrow{AB}\)
Two projectiles are launched, one at twice the speed of the other; the slower one at \(30^\circ\) and the faster one at \(60^\circ.\) Their horizontal ranges are in the ratio: (slower : faster)
| 1. | \(\dfrac{1}{2}\) | 2. | \(\dfrac{1}{4}\) |
| 3. | \(\dfrac{1}{6}\) | 4. | \(\dfrac{1}{12}\) |
In a projectile motion the velocity,
| 1. | is always perpendicular to the acceleration |
| 2. | is never perpendicular to the acceleration |
| 3. | is perpendicular to the acceleration for one instant only |
| 4. | is perpendicular to the acceleration for two instants |
Let \(\overrightarrow{\mathrm{C}}=\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}},\) then:
| 1. | \(|\overrightarrow{\mathrm{C}}|\) is always greater than \(|\overrightarrow{\mathrm{A}}|\) |
| 2. | It is possible to have \(|\overrightarrow{\mathrm{C}}|<|\overrightarrow{\mathrm{A}}|\) and \(|\overrightarrow{\mathrm{C}}|<|\overrightarrow{\mathrm{B}}|\) |
| 3. | \(|\overrightarrow{\mathrm{C}}|\) is always equal to \(|\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}|\) |
| 4. | \(|\overrightarrow{\mathrm{C}}|\) is never equal to \(|\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}|\) |
Consider the two situations shown in the figures. In figure (A), the particle \(P\) is given a velocity \(u\) up a smooth horizontal incline and it reaches a maximum vertical height \(h_A\). In figure (B), the same particle \(P\) is projected with a velocity \(u\) at an angle \(\theta\) (parallel to the previous incline) and reaches a maximum height \(h_B\).
Then,
| 1. | \(h_A=h_B~\text{sin}\theta\) |
| 2. | \(h_A~\text{sin}\theta=h_B\) |
| 3. | \(h_A~\text{sin}^2\theta=h_B\) |
| 4. | \(\frac{h_A}{\text{sin}^2\theta}=h_B\) |
| 1. | \(2\) km | 2. | \(1\) km |
| 3. | \(\dfrac12\) km | 4. | \(\dfrac14\) km |
An insect trapped in a circular groove of radius \(12\) cm moves along the groove steadily and completes \(7\) revolutions in \(100\) s. What is the angular speed of the motion?
| 1. | \(0.62\) rad/s | 2. | \(0.06\) rad/s |
| 3. | \(4.40\) rad/s | 4. | \(0.44\) rad/s |
\(\vec{r}=(3.0t\hat{i}-2.0t^{2}\hat{j}+4.0\hat{k})\) m where \(t\) is in seconds and the coefficients have the proper units for \(r\) to be in metres. The magnitude and direction of \(\vec{v}(t)\) at\(t=1.0\) s are:
1. \(4\)m/s,\(53^\circ\) with \(x\)-axis
2. \(4\) m/s, \(37^\circ\) with \(x\)-axis
3. \(5\) m/s,\(53^\circ\) with \(y\)-axis
4. \(5\)m/s, \(53^\circ\) with \(x\)-axis
Two particles having mass \(M\) and \(m\) are moving in a circular path having radius \(R\) & \(r\) respectively. If their time periods are the same, then the ratio of angular velocities will be:
1. \(\frac{r}{R}\)
2. \(\frac{R}{r}\)
3. \(1\)
4. \(\sqrt{\frac{R}{r}}\)
If two projectiles, with the same masses and with the same velocities, are thrown at an angle \(60^\circ\) & \(30^\circ\) with the horizontal, then which of the following quantities will remain the same?
| 1. | time of flight |
| 2. | horizontal range of projectile |
| 3. | maximum height acquired |
| 4. | all of the above |
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