Question

A bullet of mass 10g is fired from a gun weighing 5Kg. If the velocity of the bullet is 400m/s, what will be the speed of the recoiling gun?

Answer 1

$\underline{ \underline{\large\green\maltese \large{ \bf \red{Given:-}}}}$

$\bull \:$ $\sf Mass\; of \;the \;bullet,m_1= 10g = 0.01 kg$

$\bull \:$ $\sf Mass\; of\; the \;gun,m_2 = 5\: kg$

$\bull \:$ $\sf Final\; Velocity \;of \;the\; bullet,v_1 = 400m/s$

$\\\underline{ \underline{\large\purple\maltese \large{ \bf \orange{To\: Find:-}}}}$

$\large\dag\:$$\sf Final\; Velocity\; of \;the\; gun,v_2$

$\\\underline{ \underline{\large\blue\maltese \large{ \bf \green{Solution:-}}}}$

According to law of conservation of momentum-

$\quad\quad \pink { \underline { \boxed{ \sf{m_1u_1+m_2u_2 = m_1v_1+m_2v_2}}}}$

where

• $\sf{m_1= mass\; of \;first \;object}$
• $\sf{u_1= initial\; velocity\; of\; first\; object}$
• $\sf{m_2= mass \;of \;second \;object}$
• $\sf{u_2= initial\; velocity\; of\; second \;object}$
• $\sf{v_1= final\; velocity\; of\; first\; object}$
• $\sf{v_2= final\; velocity\; of\; second\; object}$

Since, both bodies are initially at rest, their initial velocity be zero-

$\therefore \sf{u_1 =0 }$

$\quad \sf{u_2 =0 }$

Putting Values-

$\quad\sf{0.01 × 0+5×0 = 0.01×400+5v_2}$

$\quad\sf{0 = 4+5v_2}$

$\quad\sf{5v_2= - 4}$

$\quad\sf{v_2=\dfrac{- 4}{5}m/s}$

$\quad\sf{v_2=-0.8m/s }$

$\longrightarrow$Since there is a negetive sign, the velocity of gun will be in opposite direction to that of bullet.

$\boxed{ \sf{\underline{\red{Ans}}-Speed \; of\; the \;recoiling\; gun =0.8m/s }}$

Note-

Since We have to find speed and not velocity therefore, we will not write answer with negative sign.

Answer 2

★ Given :-

• Mass of the bullet($\sf \:m_1$) = 10 g = $\sf \: \frac{10}{1000} = 0.01 \: kg$
• Mass of the gun($\sf \:m_2$) = 5 kg
• Velocity of bullet($\sf \:v_1$) = 400 m/s

⠀

★ To Find :-

• Recoil speed of the gun ($\sf \:v_2$)

⠀

★ Formula Used :-

${ \color{gold}{ \bigstar }} \: \boxed{ \sf \color{teal}m_1 u_1 + m_2 u_2 = m_1v_1 + m_2u_2}$

Where,

• $\sf \:m_1$ = mass of the bullet
• $\sf \:m_2$ = mass of the gun
• $\sf \:v_1$ = final velocity of the bullet
• $\sf \:v_2$ = final velocity of the gun
• $\sf \:u_1$ = initial velocity of the bullet
• $\sf \: u_2$ = initial velocity of the gun ⠀

Since, both these objects were at rest in the beginning. Therefore, their initial speed will be 0 m/s. Therefore, now we can do further calculations.

⠀

$\boxed{\boxed{\begin {array}{cc} \implies\color{red} \sf 0.01 \times 0 + 5 \times 0 = 0.01 \times 400 + 5 \times v \\ \implies\color{red} \sf 0 = 4 + 5v \: \: \: \: \: \: \ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \implies\color{red} \sf - 5v = 4 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\\implies\color{red} \sf v = - \frac{4}{5} =- 0.8 m/s \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \end{array}}}$

⠀

But since, it has asked us to calculate the final speed ( which is a scalar quantity) not velocity. Therefore, the final speed of the gun is $\color{tan} \boxed{\sf \: 0.8 m/s}$.

⠀

★ Concept :-

In this question, we have used the concept of conservation of momentum. It states that the sum of momentum before and after collision is equal, and it's formula is derived (Refer to formula's section for the formula). Once, we understand the question, and derive all the given information then it is easy for us to get the required answer by just putting the values in the formula and equating it, as we did in question and got our required answer, i.e. 0.8 m/s.

⠀

You can verify your answer, by putting the value of answer in it's place in the formula, just for example, we can put the value of 0.8 m/s in this question in the place of $\sf \: v_2$, and further equate it. And! if R.H.S. is equal to L.H.S. then our answer is verified.

I hope that helps you! <3