A particle starts moving with constant acceleration with initial velocity (\(\hat{\mathrm{i}}+5\hat{\mathrm{j}}\)) m/s. After \(4\) seconds, its velocity becomes (\(3\hat{\mathrm{i}}-2\hat{\mathrm{j}}\)) m/s. The magnitude of its displacement in 4 seconds is:
1. \(5\) m
2. \(10\) m
3. \(15\) m
4. \(20\) m
If a particle is moving in a circular orbit with constant speed, then:
1. | its velocity is variable. |
2. | its acceleration is variable. |
3. | its angular momentum is constant. |
4. | All of the above |
A projectile is projected with initial kinetic energy \(K\). If it has kinetic energy \(0.25K\) at its highest point, then the angle of projection is:
1. \(30^{\circ}\)
2. \(45^{\circ}\)
3. \(60^{\circ}\)
4. \(75^{\circ}\)
If the position of a particle varies according to the equations \(x= 3t^2\), \(y =2t\), and \(z= 4t+4\), then which of the following is incorrect?
1. | Velocities in \(y\) and \(z\) directions are constant |
2. | Acceleration in the \(x\text-\)direction is non-uniform |
3. | Acceleration in the \(x\text-\)direction is uniform |
4. | Motion is not in a straight line |
If three coordinates of a particle change according to the equations \(x = 3 t^{2}, y = 2 t , z= 4\), then the magnitude of the velocity of the particle at time \(t=1\) second will be:
1. \(2\sqrt{11}~\text{unit}\)
2. \(\sqrt{34}~\text{unit}\)
3. \(40~\text{unit}\)
4. \(2\sqrt{10}~\text{unit}\)
In a uniform circular motion, if the speed of the particle is \(2\) m/s and radius of the circle is \(2\) m, then the values of centripetal and tangential acceleration are, respectively:
1. \(2~\text{m/s}^2,~2~\text{m/s}^2\)
2. \(2~\text{m/s}^2,~1~\text{m/s}^2\)
3. \(0,~2~\text{m/s}^2\)
4. \(2~\text{m/s}^2,~0\)
A person, who can swim with speed \(u\) relative to water, wants to cross a river (of width \(d\) and water is flowing with speed \(v\)). The minimum time in which the person can do so is:
1. \(\frac{d}{v}\)
2. \(\frac{d}{u}\)
3. \(\frac{d}{\sqrt{v^{2} + u^{2}}}\)
4. \(\frac{d}{\sqrt{v^{2} - u^{2}}}\)
The position vector of a particle \(\overrightarrow r\) as a function of time \(t\) (in seconds) is \(\overrightarrow r=3 t \hat{i}+2t^2\hat j~\text{m}\). The initial acceleration of the particle is:
1. \(2~\text{m/s}^2\)
2. \(3~\text{m/s}^2\)
3. \(4~\text{m/s}^2\)
4. zero
Coordinates of a particle as a function of time \(t\) are \(x= 2t\),
\(y =4t\). It can be inferred that the path of the particle will be:
1. | Straight line
|
2. | Ellipse
|
3. | Parabola
|
4. | Hyperbola |
When a man walks on a horizontal road with velocity \(1\) km/h, the rain appears to him coming vertically at a speed of \(2\) km/h. The actual speed of the rain with respect to ground is:
1. \(\sqrt{3}\) km/h
2. \(\sqrt{5}\) km/h
3. \(1\) km/h
4. \(3\) km/h