Q. No.In a two-dimensional motion, instantaneous speed \(v_0\) is a positive constant. Then, which of the following is necessarily true?
1.
the average velocity is not zero at any time.
2.
average acceleration must always vanish.
3.
displacements in equal time intervals are equal.
4.
equal path lengths are traversed in equal intervals.
A car is moving at a speed of \(40\) m/s on a circular track of radius \(400\) m. This speed is increasing at the rate of \(3\) m/s2. The acceleration of the car is:
1. \(4\) m/s2
2. \(7\) m/s2
3. \(5\) m/s2
4. \(3\) m/s2
When a particle is projected at some angle to the horizontal, it has a range \(R\) and time of flight \(t_1\). If the same particle is projected with the same speed at some other angle to have the same range, its time of flight is \(t_2\), then:
1. \(t_{1} + t_{2} = \frac{2 R}{g}\)
2. \(t_{1} - t_{2} = \frac{R}{g}\)
3. \(t_{1} t_{2} = \frac{2 R}{g}\)
4. \(t_{1} t_{2} = \frac{R}{g}\)
A person reaches a point directly opposite on the other bank of a flowing river while swimming at a speed of \(5~\text{m/s}\)at an angle of \(120^\circ\) with the flow. The speed of the flow must be:
1. \(2.5~\text{m/s}\)
2. \(3~\text{m/s}\)
3. \(4~\text{m/s}\)
4. \(1.5~\text{m/s}\)
Consider the motion of the tip of the second hand of a clock. In one minute (assuming \(R\) to be the length of the second hand), its:
| 1. | displacement is \(2\pi R\) |
| 2. | distance covered is \(2R\) |
| 3. | displacement is zero. |
| 4. | distance covered is zero. |
Three girls skating on a circular ice ground of radius \(200\) m start from a point \(P\) on the edge of the ground and reach a point \(Q\) diametrically opposite to \(P\) following different paths as shown in the figure. The correct relationship among the magnitude of the displacement vector for three girls will be:
1. \(A > B > C\)
2. \(C > A > B\)
3. \(B > A > C\)
4. \(A = B = C\)
If \(\left| \vec{A}\right|\) = \(2\) and \(\left| \vec{B}\right|\) = \(4,\) then match the relations in column-I with the angle \(\theta\) between \(\vec{A}\) and \(\vec{B}\) in column-II.
| Column-I | Column-II |
| (A) \(\left| \vec{A}\times \vec{B}\right|\) \(=0\) | (p) \(\theta=30^\circ\) |
| (B)\(\left| \vec{A}\times \vec{B}\right|\)\(=8\) | (q) \(\theta=45^\circ\) |
| (C) \(\left| \vec{A}\times \vec{B}\right|\) \(=4\) | (r) \(\theta=90^\circ\) |
| (D) \(\left| \vec{A}\times \vec{B}\right|\) \(=4\sqrt2\) | (s) \(\theta=0^\circ\) |
| 1. | A(s), B(r), C(q), D(p) |
| 2. | A(s), B(p), C(r), D(q) |
| 3. | A(s), B(p), C(q), D(r) |
| 4. | A(s), B(r), C(p), D(q) |
1. \(45^{\circ} \)
2. \(90^{\circ} \)
3. \(-45^{\circ} \)
4. \(180^{\circ}\)
A cyclist starts from the center \(\mathrm{O}\) of a circular park of radius \(1\) km, reaches the edge \(\mathrm{P}\) of the park, then cycles along the circumference, and returns to the center along \(\mathrm{QO}\) as shown in the figure. If the round trip takes \(10\) min, then the average speed of the cyclist is:
1. \(22.42\) km/h
2. \(23.32\) km/h
3. \(21.42\) km/h
4. \(27.12\) km/h
A cat is situated at point \(A\) (\(0,3,4\)) and a rat is situated at point \(B\) (\(5,3,-8\)). The cat is free to move but the rat is always at rest. The minimum distance travelled by the cat to catch the rat is:
1. \(5\) unit
2. \(12\) unit
3. \(13\) unit
4. \(17\) unit
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