The displacement of a particle is proportional to the cube of the time elapsed. How does the acceleration of the particle depends on time obtained?

1. \(𝑎 ∝ t^ 2 \)
2. \(𝑎 ∝ 𝑡^ 4 \)
3. \(𝑎 ∝ 𝑡 ^3\)
4. \(𝑎 ∝ 𝑡  \)

Starting from rest, the acceleration of a particle is \(𝑎 = 2 ( 𝑡 − 1 )\). The velocity of the particle at \(𝑡 = 5\ \text{𝑠}\) is:

1.  \(15\ \text{m/s}\)
2.  \(25\ \text{m/s}\)
3.  \(5\ \text{m/s}\)
4.  None of these

The speeds of two identical cars are \(u\) and \(4u\) at a specific instant. The ratio of the respective distances in which the two cars are stopped from that instant is:

1. \(1 : 1\)
2. \(1 : 4\)
3. \(1 : 8\)
4. \(1 : 16\)

A body is moving with uniform acceleration describes \(40\ \text{m}\) in the first \(5\) seconds and \(65\ \text{m}\) in the next  \(5\) seconds. Its initial velocity will be:

1. \(4\ \text{m/s}\)

2. \(2.5\ \text{m/s}\)

3. \(5.5\ \text{m/s}\)

4. \(11\ \text{m/s}\)

The displacement \(x\) of a particle varies with time \(𝑡\) as \(x=ae^{\alpha t}+be^{\beta t}\) where \(𝑎,\ 𝑏,\ 𝛼\) and \(\beta\) are positive constants. The velocity of the particle will:

1. Go on decreasing with time
2. Be independent of \(𝛼\) and \(\beta\)
3. Drop to zero when \(𝛼 = 𝛽\)
4. Go on increasing with time

A car, starting from rest, accelerates at the rate \(f\) through a distance \(S\), then continues at a constant speed for time \(t\) and then decelerates at the rate \(\frac f2\) to come to rest. If the total distance traversed is \(15\ \text{S}\), then:

1. \(S = \frac{1}{2}ft^2\)
2. \(S = \frac{1}{4}ft^2\)
3. \(S = \frac{1}{72}ft^2\)
4. \(S = \frac{1}{6}ft^2\)

A man is \(45\ \text{m}\) behind the bus when the bus starts accelerating from rest with an acceleration of \(2.5\ \text{m/s}^2\). With what minimum velocity should the man start running to catch the bus?

1.  \(12\ \text{m/s}\)
2.  \(14\ \text{m/s}\)
3.  \(15\ \text{m/s}\)
4.  \(16\ \text{m/s}\)

A particle moves along the x-axis as \({x}=4({t}-2)+{a}({t}-2)^2.\)Which of the following is true? 

A body starting from rest moves with constant acceleration. The ratio of the distance covered by the body during the \(5^{th}\) second to that covered in \(5\) second is:

1. \(9/25\)
2. \(3/5\)
3. \(25/9\)
4. \(1/25\)

Two trains, each \(50\) m long, are travelling in the opposite direction with velocities \(10\) m/s and \(15\) m/s. The time of crossing is:
1. \(10\) sec
2. \(4\) sec
3. \(2\sqrt{3}\) sec
4. \(4\sqrt{3}\) sec