If a body travels some distance in a given time interval, then for that time interval, its:

A car moves from \(\mathrm{X}\) to \(\mathrm{Y}\) with a uniform speed \(\mathrm{v_u}\) and returns to \(\mathrm{X}\) with a uniform speed \(\mathrm{v_d}.\) The average speed for this round trip is:

1. $\frac{2{\mathrm{v}}_{\mathrm{d}}{\mathrm{v}}_{\mathrm{u}}}{{\mathrm{v}}_{\mathrm{d}}+{\mathrm{v}}_{\mathrm{u}}}$

2. $\sqrt{{\mathrm{v}}_{\mathrm{u}}{\mathrm{v}}_{\mathrm{d}}}$

3. $\frac{{\mathrm{v}}_{\mathrm{d}}{\mathrm{v}}_{\mathrm{u}}}{{\mathrm{v}}_{\mathrm{d}}+{\mathrm{v}}_{\mathrm{u}}}$

4. $\frac{{\mathrm{v}}_{\mathrm{u}}+{\mathrm{v}}_{\mathrm{d}}}{2}$

The figure gives the \((\mathrm{x-t})\) plot of a particle in a one-dimensional motion. Three different equal intervals of time are shown. The signs of average velocity for each of the intervals \(1,\) \(2\) & \(3,\) respectively are:

A vehicle travels half the distance \(\mathrm{L} \) with speed \(\mathrm{v_1}\) and the other half with speed \(\mathrm{v_2},\) then its average speed is:
1. $\frac{{\mathrm{v}}_1+{\mathrm{v}}_2}{2}$

2. $\frac{2{\mathrm{v}}_1+{\mathrm{v}}_2}{{\mathrm{v}}_1+{\mathrm{v}}_2}$

3. $\frac{2{\mathrm{v}}_1{\mathrm{v}}_2}{{\mathrm{v}}_1+{\mathrm{v}}_2}$

4. $\frac{\mathrm{L}({\mathrm{v}}_1+{\mathrm{v}}_2)}{{\mathrm{v}}_1{\mathrm{v}}_2}$

The coordinate of an object is given as a function of time by $\mathrm{x}=7\mathrm{t}-3{\mathrm{t}}^2$, where x is in metres and t is in seconds. Its average velocity over the interval t=0 to t=4 is will be:

1.  5 m/s

2.  -5 m/s

3.  11 m/s

4.  -11 m/s

A particle moving in a straight line covers half the distance with a speed of \(3~\text{m/s}\). The other half of the distance is covered in two equal time intervals with speeds of \(4.5~\text{m/s}\) and \(7.5~\text{m/s}\) respectively. The average speed of the particle during this motion is:
1. \(4.0~\text{m/s}\)
2. \(5.0~\text{m/s}\)
3. \(5.5~\text{m/s}\)
4. \(4.8~\text{m/s}\)

A car is moving along a straight line, say OP in the figure. It moves from O to P in 18 s and returns from P to Q in 6.0 s. The average velocity and average speed of the car in going from O to P and back to Q respectively are:

1.  10 m/s & 10 m/s

2.  20 m/s & 30 m/s 

3.  20 m/s & 20 m/s

4.  10 m/s & 20 m/s

A particle moves with a velocity $\mathrm{v}={\mathrm{\alpha t}}^3$ along a straight line. The average speed in time interval \(t=0\) to \(t=T\) will be:

1. ${\mathrm{\alpha T}}^3$

2. $\frac{{\mathrm{\alpha T}}^3}{2}$

3. $\frac{{\mathrm{\alpha T}}^3}{3}$

4. $\frac{{\mathrm{\alpha T}}^3}{4}$

The position of an object moving along \(\mathrm{x}\)-axis is given by \(x=a+bt^2\), where \(a=8.5\) m, \(b=2.5\) ${\mathrm{ms}}^{-2}$ and \(t\) is measured in seconds. Its average velocity between \(t=2.0\) s and \(t=4.0\) s is:
1. \(10\) m/s
2. \(15\) m/s
3. \(20\) m/s
4. \(25\) m/s

A particle starts from rest with constant acceleration. The ratio of space-average velocity to the time-average velocity is:

where time-average velocity and space-average velocity, respectively, are defined as follows: 

$<v{>}_{time}$ $=$ $\frac{\int vdt}{\int dt}$
$<v{>}_{space}$ $=$ $\frac{\int vds}{\int ds}$ $$

1. $\frac{1}{2}$

2. $\frac{3}{4}$

3. $\frac{4}{3}$

4. $\frac{3}{2}$