Two particles \(\mathrm{A}\) and \(\mathrm{B}\), move with constant velocities \(\overrightarrow{{v}_1}\) and \(\overrightarrow{{v}_2}\) respectively. At the initial moment, their position vectors are \(\overrightarrow{{r}_1}\) and \(\overrightarrow{{r}_2}\) respectively. The condition for particles \(\mathrm{A}\) and \(\mathrm{B}\) for their collision will be:
1.\(\dfrac{\vec{r_1}-\vec{r_2}}{\left|\vec{r_1}-\vec{r_2}\right|}=\dfrac{\vec{v_2}-\vec{v_1}}{\left|\vec{v_2}-\vec{v_1}\right|}\)
2. \(\vec{r_1} \cdot \vec{v_1}=\vec{r_2} \cdot \vec{v_2}\)
3. \(\vec{r_1} \times \vec{v_1}=\vec{r_2} \times \vec{v_2}\)
4. \(\vec{r_1}-\vec{r_2}=\vec{v_1}-\vec{v_2}\)
The position vector of a particle \(\vec{R}\) as a function of time \(t\) is given by;
\(\overrightarrow{\mathrm{R}}=4 \sin (2 \pi \mathrm{t}) \hat{\mathrm{i}}+4 \cos (2 \pi \mathrm{t}) \hat{\mathrm{j}}\)
Where \(R\) is in meters, \(t\) is in seconds and \(\mathrm{\hat{i},\hat{j}}\) denotes unit vectors along \(\mathrm{x}\) and \(\mathrm{y}\)-directions, respectively. Which one of the following statements is wrong for the motion of the particle?
1. | acceleration is along \(-\overrightarrow{R}\). |
2. | magnitude of the acceleration vector is \(\frac{v^2}{R}\), where \(v\) is the velocity of the particle. |
3. | magnitude of the velocity of the particle is \(8\) m/s. |
4. | path of the particle is a circle of radius \(4\) m. |
The velocity of a projectile at the initial point \(A\) is \(2\hat i+3\hat j~\)m/s. Its velocity (in m/s) at point \(B\) is:
1. | \(-2\hat i+3\hat j~\) | 2. | \(2\hat i-3\hat j~\) |
3. | \(2\hat i+3\hat j~\) | 4. | \(-2\hat i-3\hat j~\) |
A particle moves in a circle of radius \(5\) cm with constant speed and time period \(0.2\pi\) s. The acceleration of the particle is:
1. | \(25\) m/s2 | 2. | \(36\) m/s2 |
3. | \(5\) m/s2 | 4. | \(15\) m/s2 |
A missile is fired for a maximum range with an initial velocity of 20 m/s. If g= 10 m/s2, then the range of the missile will be:
1. 50 m
2. 60 m
3. 20 m
4. 40 m
A particle of mass m is projected with velocity v making an angle of 45° with the horizontal. When the particle lands on level ground, the magnitude of change in its momentum will be:
1.
2.
3.
4. zero
A particle starting from the origin \((0,0)\) moves in a straight line in the \((x,y)\) plane. Its coordinates at a later time are (, \(3\)). The path of the particle makes an angle of __________ with the x-axis:
1. \(30^\circ\)
2. \(45^\circ\)
3. \(60^\circ\)
4. \(0\)
For a projectile projected at angles (45°-θ) and (45°+θ), the horizontal ranges described by the projectile are in the ratio of:
1. 1:1
2. 2:3
3. 1:2
4. 2:1
A car turns at a constant speed on a circular track of radius \(100\) m, taking \(62.8\) s for every circular lap. The average velocity and average speed for each circular lap, respectively, is:
1. | \(0,~0\) | 2. | \(0,~10\) m/s |
3. | \(10\) m/s, \(10\) m/s | 4. | \(10\) m/s, \(0\) |