The kinetic energy of a person is just half of the kinetic energy of a boy whose mass is just half of that person. If the person increases his speed by \(1~\text{m/s},\) then his ​​​​​​​kinetic energy equals to that of the boy, then the initial speed of the person was:
1. \(\left( \sqrt{2}+1 \right)~\text{m/s}\)
2. \(\left( 2+\sqrt{2} \right)~\text{m/s}\)
3. \(2\left( 2+\sqrt{2} \right)~\text{m/s}\)
4. none of the above

A particle of mass m₁ is moving with a velocity v₁ and another particle of mass m₂ is moving with a velocity v₂. Both of them have the same momentum, but their kinetic energies are E₁ and E₂ respectively. If m₁ > m₂ then: 

1. $\frac{{\mathrm{E}}_1}{{\mathrm{E}}_2}=\frac{{\mathrm{m}}_1}{{\mathrm{m}}_2}$

2. ${\mathrm{E}}_1>{\mathrm{E}}_2$

3. ${\mathrm{E}}_1={\mathrm{E}}_2$

4. ${\mathrm{E}}_1<{\mathrm{E}}_2$

A stone is tied to a string of length 'l' is whirled in a vertical circle with the other end of the string as the centre. At a certain instant of time, the stone is at its lowest position and has a speed 'u'. The magnitude of the change in velocity as it reaches a position where the string is horizontal (g being acceleration due to gravity) is:

1. $\sqrt{u^2-g\ell }$

2. $u-\sqrt{u^2-2g\ell }$

3. $\sqrt{2\mathrm{g}\ell }$

4. $\sqrt{2({\mathrm{u}}^2-\mathrm{g}\ell )}$

A mass of \(0.5~\text{kg}\) moving with a speed of \(1.5~\text{m/s}\) on a horizontal smooth surface, collides with a nearly weightless spring with force constant \(k=50~\text{N/m}.\) The maximum compression of the spring would be:
                   
1.  \(0.12~\text{m}\) 
2.  \(1.5~\text{m}\) 
3.  \(0.5~\text{m}\) 
4.  \(0.15~\text{m}\) 

A particle projected with velocity 'u' makes an angle θ with respect to horizontal. Now it breaks in two identical parts at highest point of trajectory. If one part retraces its path, then velocity of the other part is:
1. 3u cos θ
2. 2u cos θ
3. u cos θ
4. u

If two springs, A and B $(K_A$ $=$ $2$ $K_B),$ are stretched by the same suspended weights, then the ratio of work done in stretching is equal to:
1.  1 : 2
2.  2 : 1
3.  1 : 1
4.  1 : 4

When a spring is subjected to 4 N force, its length is a metre and if 5 N is applied, its length is b metre. If 9 N is applied, its length will be:

1.  4b – 3a

2.  5b – a

3.  5b – 4a

4.  5b – 2a

If the kinetic energy of a body is increased by
\(300\)%, then the percentage change in momentum will be:
1. \(100\)%
2. \(150\)%
3. \(265\)%
4. \(73.2\)%

A child is sitting on a swing. Its minimum and maximum heights from the ground are \(0.75\) m and \(2\) m, respectively. Its maximum speed will be: (take \(g=10\) m/s²)
1. \(10\) m/s
2. \(5\) m/s
3. \(8\) m/s
4. \(15\) m/s

Two springs A and B having spring constant $K_A$ $and$ $K_B$ $$ $(K_A$ $=$ $2K_B)$ are stretched by applying a force of equal magnitude. If the energy stored in spring A is E, then the energy stored in B will be:

1.  2E

2.  $\frac{E}{4}$

3. $\frac{E}{2}$

4.  4E