Two particles are separated by a horizontal distance \(x\) as shown in the figure. They are projected at the same time as shown in the figure with different initial speeds. The time after which the horizontal distance between them becomes zero will be:
1. | \(\frac{x}{u}\) | 2. | \(\frac{u}{2 x}\) |
3. | \(\frac{2 u}{x}\) | 4. | None of the above |
A particle starts from the origin at \(t=0\) with a velocity of \(5.0\hat i\) m/s and moves in the \(x\text-y\) plane under the action of a force that produces a constant acceleration of \((3.0\hat i + 2.0\hat j)~\text{m/s}^2\). What is the speed of the particle at the instant its \(x\text-\)coordinate is \(84\) m?
1. | \(36\) m/s | 2. | \(26\) m/s |
3. | \(1\) m/s | 4. | \(0\) m/s |
Two boys are standing at the ends \(A\) and \(B\) of the ground where \(AB =a.\) The boy at \(B\) starts running in a direction perpendicular to \(AB\) with velocity \(v_1.\) The boy at \(A\) starts running simultaneously with velocity \(v\) and catches the other boy in a time \(t,\) where \(t\) is:
1. | \(\frac{a}{\sqrt{v^2+v^2_1}}\) | 2. | \(\frac{a}{\sqrt{v^2-v^2_1}}\) |
3. | \(\frac{a}{v-v_1}\) | 4. | \(\frac{a}{v+v_1}\) |
A stone tied to the end of a \(1\) m long string is whirled in a horizontal circle at a constant speed. If the stone makes \(22\) revolutions in \(44\) seconds, what is the magnitude and direction of acceleration of the stone?
1. | \(\pi^2 ~\text{ms}^{-2} \) and direction along the tangent to the circle. |
2. | \(\pi^2 ~\text{ms}^{-2} \) and direction along the radius towards the centre. |
3. | \(\frac{\pi^2}{4}~\text{ms}^{-2} \) and direction along the radius towards the centre. |
4. | \(\pi^2~\text{ms}^{-2} \) and direction along the radius away from the centre. |
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 \(5\) m/s at an angle of \(120^\circ\) with the flow. The speed of the flow must be:
1. \(2.5\) m/s
2. \(3\) m/s
3. \(4\) m/s
4. \(1.5\) 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. |
A particle starts from the origin at \(t=0\) sec with a velocity of \(10\hat j~\text{m/s}\) and moves in the \(x\text-y\) plane with a constant acceleration of \((8.0\hat i +2.0 \hat j)~\text{m/s}^2\). At what time is the \(x\text-\)coordinate of the particle \(16~\text{m}\)?
1. \(2\) s
2. \(3\) s
3. \(4\) s
4. \(1\) s