A particle moving in a circle of radius \(R\) with a uniform speed takes a time \(T\) to complete one revolution. If this particle were projected with the same speed at an angle \(\theta\) to the horizontal, the maximum height attained by it equals \(4R.\) The angle of projection, \(\theta\) is then given by:
1. \( \theta=\sin ^{-1}\left(\frac{\pi^2 {R}}{{gT}^2}\right)^{1/2}\)
2. \(\theta=\sin ^{-1}\left(\frac{2 {gT}^2}{\pi^2 {R}}\right)^{1 / 2}\)
3. \(\theta=\cos ^{-1}\left(\frac{{gT}^2}{\pi^2 {R}}\right)^{1 / 2}\)
4. \(\theta=\cos ^{-1}\left(\frac{\pi^2 {R}}{{gT}^2}\right)^{1 / 2}\)
A car starts from rest and accelerates at . At \(t=4\) s, a ball is dropped out of a window by a person sitting in the car. What is the velocity and acceleration of the ball at \(t=6\) s? (Take \(g=10\) m/s2)
1.
2.
3.
4.
Rain is falling vertically downward with a speed of \(35~\text{m/s}\). Wind starts blowing after some time with a speed of \(12~\text{m/s}\) in East to West direction. The direction in which a boy standing at the place should hold his umbrella is:
1. | \(\text{tan}^{-1}\Big(\frac{12}{37}\Big)\) with respect to rain |
2. | \(\text{tan}^{-1}\Big(\frac{12}{37}\Big)\) with respect to wind |
3. | \(\text{tan}^{-1}\Big(\frac{12}{35}\Big)\) with respect to rain |
4. | \(\text{tan}^{-1}\Big(\frac{12}{35}\Big)\) with respect to wind |
1. | \(3000\) m | 2. | \(2800\) m |
3. | \(2000\) m | 4. | \(1000\) m |
1. | \(4\sqrt2~\text{ms}^{-1},45^\circ\) | 2. | \(4\sqrt2~\text{ms}^{-1},60^\circ\) |
3. | \(3\sqrt2~\text{ms}^{-1},30^\circ\) | 4. | \(3\sqrt2~\text{ms}^{-1},45^\circ\) |
1. | \(20\) | 2. | \(10\sqrt3\) |
3. | zero | 4. | \(10\) |
1. | \(\vec v\) is a constant; \(\vec a\) is not a constant |
2. | \(\vec v\) is not a constant; \(\vec a\) is not a constant |
3. | \(\vec v\) is a constant; \(\vec a\) is a constant |
4. | \(\vec v\) is not a constant; \(\vec a\) is a constant |
Two bullets are fired horizontally and simultaneously towards each other from the rooftops of two buildings (building being \(100~\text{m}\) apart and being of the same height of \(200~\text{m}\)) with the same velocity of \(25~\text{m/s}\). When and where will the two bullets collide? \(\left(g = 10~\text{m/s}^2 \right)\)?
1. | after \(2~\text{s}\) at a height of \(180~\text{m}\) |
2. | after \(2~\text{s}\) at a height of \(20~\text{m}\) |
3. | after \(4~\text{s}\) at a height of \(120~\text{m}\) |
4. | they will not collide. |