Question

A rubber cord catapult has a cross-sectional area 25 \[m{{m}^{2}}\] and initial length of rubber cord is 10 cm. It is stretched to 5 cm and then released to project a missile of mass 5g. Taking Y rubber=\[Y=5\times {{10}^{8}}N{{m}^{-2}}\], velocity of projected missile is:
A.20 \[m{{s}^{-1}}\]
B.100 \[m{{s}^{-1}}\]
C.250 \[m{{s}^{-1}}\]
D.200 \[m{{s}^{-1}}\]

Answer 1

Hint: The energy conservation is taking place here. Energy conservation tell that neither energy can be created nor be destroyed .The potential energy stored in the rubber cord will be converted into kinetic energy.

Complete step by step answer:
The details given in the question are:
Cross sectional area is 25 \[m{{m}^{2}}\].
Initial length of rubber cord is 10 cm.
Then it is stretched to 5cm.
Mass of missile projected=5g.
We know that potential energy of rubber cord catapult is given as,

$\begin{align}
  & U=\dfrac{1}{2}\dfrac{YA{{l}^{2}}}{L} \\
 & \\
\end{align}$

Where A is the area of cross section, l is the stretched length l is the initial length of the cord, and Y is the young’s modulus of wire given as
\[Y=5\times {{10}^{8}}N{{m}^{-2}}\]
  And the kinetic energy of the mass is given by

$V=\dfrac{1}{2}m{{v}^{2}}$
Where m is the mass of the missile
v is the velocity of the missile
According to energy conservation,
U=V
Therefore,

$\begin{align}
  & \dfrac{1}{2}m{{v}^{2}}=\dfrac{1}{2}\dfrac{YA{{l}^{2}}}{L} \\
 & v=\sqrt{\dfrac{YA{{l}^{2}}}{mL}} \\
 & v={{\sqrt{\dfrac{5\times {{10}^{8}}\times 25\times {{10}^{-6}}\times \left( 5\times {{10}^{-2}} \right)}{5\times {{10}^{-3}}\times 10\times {{10}^{-2}}}}}^{2}} \\
\end{align}$
So
$v=250$\[m{{s}^{-1}}\]
Therefore the required solution is option C.

Note:
The equation for potential energy can be misunderstood here .There are a lot of equations for potential energy and kinetic energy. Therefore it is to be importantly noted that the potential energy as well as kinetic energy formula must have link with situations mentioned in the question. The values of other quantities mentioned in the question may help us to do this. Young's modulus, also known as the Young modulus, is a mechanical characteristic that measures the stiffness of a solid material. It explains the relationship between stress and strain in a material in the linear elasticity regime of a uniaxial deformation.