A particle moves along \(x\)-axis in such a way that its coordinate \(x\) varies with time \(t\) according to the equation \(𝑥 = ( 2 − 5 𝑡 + 6 𝑡^ 2 )  \) m. The initial velocity of the particle is:

1. \(–5\ \text{m/s}\)

2. \(6\ \text{m/s}\)

3. \(–3\ \text{m/s}\)

4. \(3\ \text{m/s}\)

A car starts from rest and moves with uniform acceleration \(a\) on a straight road from time \(t = 0\) to \(t = T\). After that, a constant deceleration brings it to rest. In this process, the average speed of the car is: 

1. \(\dfrac{aT}{4}\)

2. \(\dfrac{3aT}{2}\)

3. \(\dfrac{aT}{2}\)

4. \(aT\)

An object accelerates from rest to a velocity of \(27.5\ \text{m/s}\) in \(10\ \text{s}\). Then find the distance covered by the object in the next \(10\ \text{s}\):

1. \(550\ \text{m}\)
2. \(137.5\ \text{m}\)
3. \(412.5\ \text{m}\)
4. \(275\ \text{m}\)

If the velocity of a particle is given by \(v = (180-16x)^{1/2}~\text{m/s} \), then its acceleration will be: 
1. zero
2. \(8\text{ m/s}^2\)
3. \(-8\text{ m/s}^2\)
4. \(4\text{ m/s}^2\)

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

Speed 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 m in the first 5 sec and 65 m in the next 5 sec. Its initial velocity will be:

1. 4 m/s

2. 2.5 m/s

3. 5.5 m/s

4. 11 m/s 

The displacement x of a particle varies with time $t,<mspace\; width="1em"></mspace>x=a{e}^{-\alpha t}+b{e}^{\beta t}$, where $a,b,\alpha \text{\hspace{0.17em}and\hspace{0.17em}}\beta$ are positive constants. The velocity of the particle will 

1. Go on decreasing with time
2. Be independent of $\alpha$ and $\beta$
3. Drop to zero when $\alpha$ = $\beta$
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 $\frac{f}{2}$ to come to rest. If the total distance traversed is 15 S, then, 

1. $S=\frac{1}{2}f{t}^2$

2. $S=\frac{1}{4}f{t}^2$

3. $S=\frac{1}{72}f{t}^2$  

4. $S=\frac{1}{6}f{t}^2$