Question

A bullet of mass 10g moving horizontally with a velocity of 400$m{{s}^{-1}}$ strikes a wood block of mass 2kg which is suspended by light inextensible string of length 5m. As a result, the centre of gravity of the block was found to rise a vertical distance of 10cm. The speed of the bullet after it emerges horizontally from the block will be:
$\text{A}\text{. }100m{{s}^{-1}}$
$\text{B}\text{. 8}0m{{s}^{-1}}$
$\text{C}\text{. 12}0m{{s}^{-1}}$
$\text{D}\text{. 16}0m{{s}^{-1}}$

Answer 1

Hint: It is given that the block raises to a height of 10cm. Use work energy theorem, i.e. $W=\Delta K$ to find the horizontal velocity of the block when the bullet emerges out of it. Then use the law of conservation of momentum to find the horizontal velocity of the bullet when it emerges out of the block.

Formula used:
$W=\Delta K$
${{P}_{i}}={{P}_{f}}$
p=mv
$K=\dfrac{1}{2}Mv_{1}^{2}$

Complete step by step answer:
Let's consider the value of acceleration due to gravity equal to 10$m{{s}^{-2}}$, for this question.
It is given that a bullet with a horizontal velocity of 400$m{{s}^{-1}}$ strikes a block of 2kg. It is given that the bullet passes through the block and we have to find the horizontal velocity of the bullet when it emerges out of the block.
To find this horizontal velocity, it is also given that after the bullet passes through the block, the block gains some momentum and the centre of gravity of the block rises to a height of 10cm.
Let us understand the case more clearly.
The bullet comes with some velocity and strikes the block and emerges out of the block. In this process, the bullet exerts a force on the block and the block gains some velocity. Let this velocity of the block be ${{v}_{1}}$.
Since the block is suspended to a string, it will undergo a circular motion. However, gravity will oppose its motion and after some time it comes to rest at a height of h=10cm. Therefore, gravity does a negative work on the block, which is equal to W = -Mgh,
where M is the mass of the block, g is acceleration due to gravity.
Convert 10 cm to metres.
$\Rightarrow 10cm=10\times {{10}^{-2}}m=0.1m$
Hence, $W=-2\times 10\times 0.1=-2J$.
According to the work energy theorem, work done on a body is equal to the change in its kinetic energy. i.e. $W=\Delta K$.
Here, $\Delta K=0-\dfrac{1}{2}Mv_{1}^{2}=-\dfrac{1}{2}\times 2v_{1}^{2}=-v_{1}^{2}$.
Hence, $-v_{1}^{2}=-2$
${{v}_{1}}=\sqrt{2}m{{s}^{-1}}$.
This means that when the bullet emerges out of the block, the horizontal velocity of the block is ${{v}_{1}}=\sqrt{2}m{{s}^{-1}}$.
When the bullet passes through the block, there are only two forces acting on the system and that are the equal and opposite forces exerted by both the bodies on each other. Hence, the net force on the system is zero. Therefore, we can use the law of conservation of momentum that says that when the net force is zero, the final and the initial momentums are equal, i.e. ${{P}_{i}}={{P}_{f}}$.
The formula for momentum is p=mv, where m is the mass of the moving body and v is its velocity.
Let the horizontal velocity of the bullet when it emerges out be ${{v}_{2}}$.
Here, ${{P}_{i}}=10g\times 400m{{s}^{-1}}=10\times {{10}^{-3}}kg\times 400m{{s}^{-1}}=4kgm{{s}^{-1}}$.
And ${{P}_{f}}=2\times \sqrt{2}+10\times {{10}^{-3}}\times {{v}_{2}}$.
$\Rightarrow {{P}_{f}}=2\sqrt{2}+0.01{{v}_{2}}$
And ${{P}_{i}}={{P}_{f}}$.
$\Rightarrow 4=2\sqrt{2}+0.01{{v}_{2}}$
$\Rightarrow 0.01{{v}_{2}}=4-2\sqrt{2}$
$\sqrt{2}\approx 1.4$
$\Rightarrow 0.01{{v}_{2}}=4-2\times 1.4$
$\Rightarrow 0.01{{v}_{2}}=4-2.8=1.2$
$\Rightarrow {{v}_{2}}=\dfrac{1.2}{0.01}=120m{{s}^{-1}}$.
This means the horizontal velocity of the bullet when it emerges out of the bullet is $120m{{s}^{-1}}$.
Hence, the correct option is C.

Note: Always remember that the units of all the physical quantities involved in the given question must be in the same system of units. Mainly, it is the system of SI units.
For example, all the units are SI units except the units of mass of the bullet and height. The mass of the bullet is given in grams. So we have to convert its unit in SI unit i.e. kilogram(kg).