A particle moves in space such that:
\(x=2t^3+3t+4;~y=t^2+4t-1;~z=2\sin\pi t\)
where \(x,~y,~z\) are measured in meters and \(t\) in seconds. The acceleration of the particle at \(t=3\) seconds will be:
1. | \(36 \hat{i}+2 \hat{j}+\hat{k} \) ms-2 |
2. | \(36 \hat{i}+2 \hat{j}+\pi \hat{k} \) ms-2 |
3. | \(36 \hat{i}+2 \hat{j} \) ms-2 |
4. | \(12 \hat{i}+2 \hat{j} \) ms-2 |
A projectile is fired from the surface of the earth with a velocity of \(5\) ms–1 and at an angle \(\theta\) with the horizontal. Another projectile fired from another planet with a velocity of \(3\) ms–1 at the same angle follows a trajectory that is identical to the trajectory of the projectile fired from the Earth. The value of the acceleration due to gravity on the other planet is: (given \(g=9.8\) ms–2)
1. \(3.5\) m/s2
2. \(5.9\) m/s2
3. \(16.3\) m/s2
4. \(110.8\) m/s2
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 body is moving with a velocity of \(30\) m/s towards the east. After \(10\) s, its velocity becomes \(40\) m/s towards the north. The average acceleration of the body is:
1. \( 7~\text{m/s}^2\)
2. \( \sqrt{7}~\text{m/s}^2\)
3. \(5~\text{m/s}^2\)
4. \(1~\text{m/s}^2\)
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 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 \((45^{\circ}-\theta)\) and \((45^{\circ}+\theta)\), 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, \(0\) |