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. |
In a two-dimensional motion, instantaneous speed \(v_0\) is a positive constant. Then which of the following is necessarily true?
1. | The acceleration of the particle is zero. |
2. | The acceleration of the particle is increasing. |
3. | The acceleration of the particle is necessarily in the plane of motion. |
4. | The particle must be undergoing a uniform circular motion. |
The speed of a projectile at its maximum height is half of its initial speed. The angle of projection is:
1. \(15^{\circ}\)
2. \(30^{\circ}\)
3. \(45^{\circ}\)
4. \(60^{\circ}\)
A particle moves in the \((x\text-y)\) plane according to the rule \(x = a \sin (\omega t)\) and \(y = a \cos (\omega t)\). The particle follows:
1. | a circular path. |
2. | a parabolic path. |
3. | a straight line path inclined equally to x and y-axes. |
4. | an elliptical path. |
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. \(t_{1} t_{2} = \frac{R}{g}\)
The equation of a projectile is \(y = ax -bx^{2}\). Its horizontal range is?
1. \(\frac{a}{b}\)
2. \(\frac{b}{a}\)
3. \(a+b\)
4. \(b-a\)
The velocity at the maximum height of a projectile is \(\frac{\sqrt{3}}{2}\) times its initial velocity of projection \((u)\). Its range on the horizontal plane is:
1. \(\frac{\sqrt{3} u^{2}}{2 g}\)
2. \(\frac{3 u^{2}}{2 g}\)
3. \(\frac{3 u^{2}}{ g}\)
4. \(\frac{u^{2}}{2 g}\)
The position coordinates of a projectile projected from ground on a certain planet (with no atmosphere) are given by
\(y =4 t - 2 t^{2}~ \text{m}\) and \(x =3t\) metre, where \(t\) is in seconds and point of projection is taken as the origin. The angle of projection of projectile with vertical is:
1. \(30^{\circ}\)
2. \(37^{\circ}\)
3. \(45^{\circ}\)
4. \(60^{\circ}\)
A particle projected from origin moves in the \(x\text-y\) plane with a velocity \(\overrightarrow{v} = 3 \hat{i} + 6 x \hat{j}\), where \(\hat i\) and \(\hat j\) are the unit vectors along the \(x\) and \(y\text-\)axis. The equation of path followed by the particle is:
1. \(y=x^2\)
2. \(y=\frac{1}{x^2}\)
3. \(y=2x^2\)
4. \(y=\frac{1}{x}\)