The position of a particle moving in the XY plane at any time \(t\) is given by \(𝑥 = ( 3 𝑡^ 2 − 6 𝑡 )\) metres. Select the correct statement about the moving particle from the following.

1. The acceleration of the particle is zero at \(t = 0\) second
2. The velocity of the particle is zero at \(t = 0\) second
3. The velocity of the particle is zero at \(t = 1\) second
4. The velocity and acceleration of the particle are never zero

If a body having initial velocity zero is moving with uniform acceleration \(8\ \text{m/s}^2\), then the distance travelled by it in the fifth second will be: 

1. \(36\) metres

2. \(40\) metres

3. \(100\) metres

4. Zero

An alpha particle enters a hollow tube of \(4\ \text{m}\) length with an initial speed of \(1\ \text{km/s}\). It is accelerated in the tube and comes out of it with a speed of \(9\ \text{km/s}\). The time for which it remains inside the tube is:

1. \(8 × 10^{ − 3} \ \text{s}\)

2. \(80 × 10^{ − 3} \ \text{s}\)$\mathrm{ }$

3. \(800 × 10^{ − 3} \ \text{s}\)$\mathrm{}$

4. \(8 × 10^{ − 4} \ \text{s}\)$\mathrm{}$

Two cars \(A\) and \(B\) are travelling in the same direction with velocities \(v_1\) and \(v_2 (v_1>v_2)\). When the car \(A\) is at a distance \(d\) behind car \(B\), the driver of the car \(A\) applied the brake producing uniform retardation \(a\). There will be no collision when:
1. \(d< \dfrac{(v_1-v_2)^2}{2a}\)

2. \(d< \dfrac{v^2_1-v^2_2}{2a}\)

3. \(d> \dfrac{(v_1-v_2)^2}{2a}\)

4. \(d> \dfrac{v^2_1-v^2_2}{2a}\)

A body of mass \(10\ \text{kg}\) is moving with a constant velocity of \(10\ \text{m/s}\). When a constant force acts for \(4\ \text{s}\) on it, it moves with a velocity \(2\ \text{m/s}\) in the opposite direction. The acceleration produced in it is:

1. \(3\ \text{m/s}^2\)
2. \(-3\ \text{m/s}^2\)
3. \(0.3\ \text{m/s}^2\)
4. \(-0.3\ \text{m/s}^2\)

A body starts from rest from the origin with an acceleration of \(6~\text{m/s}^2\) along the \(x\text-\)axis and \(8~\text{m/s}^2\) along the \(y\text-\)axis. Its distance from the origin after \(4\) seconds will be:
1. \(56~\text{m}\)
2. \(64~\text{m}\)
3. \(80~\text{m}\)
4. \(128~\text{m}\)

A car moving with a velocity of \(10\ \text{m/s}\) can be stopped by the application of a constant force \(F\) in a distance of \(20\ \text{m}\). If the velocity of the car is \(30\ \text{m/s}\), it can be stopped by this force in:

1. \(\dfrac {20}{3} \ \text{𝑚}\)

2. \(20\ \text{m}\)

3. \(60\ \text{m}\)

4. \(180\ \text{m}\)

The displacement of a particle is given by \(y = a + bt + ct^{2} - dt^{4}\). The initial velocity and acceleration are, respectively:

A car moving with a speed of \(40\ \text{km/h}\) can be stopped by applying the brakes for at least \(2\ \text{m}\). If the same car is moving with a speed of \(80\ \text{km/h}\), what is the minimum stopping distance?

1. \(8\ \text{m}\)
2. \(2\ \text{m}\)
3. \(4\ \text{m}\)
4. \(6\ \text{m}\)

An elevator car, whose floor-to-ceiling distance is equal to \(2.7~\text{m}\), starts ascending with constant acceleration of \(1.2~\text{ms}^{-2}\). \(2\ \text{s}\)  after the start, a bolt begins falling from the ceiling of the car. The free-fall time of the bolt is: 
1. \(\sqrt{0.54}~\text{s}\)
2. \(\sqrt{6}~\text{s}\)
3. \(0.7~\text{s}\)
4. \(1~\text{s}\)