Question

A bullet leaving the muzzle of a rifle barrel with a

velocity v penetrates a plank and loses one fifth of its

velocity. It then strikes second plank, which it just

penetrates through. Find the ratio of the thickness of

the planks supposing average resistance to the

penetration is same in both the cases.

Answer 1

Velocity of bullet after the first plank will be,

=> $v^{2} = u^{2} - 2a(s1)$

Now we know,

=> v = (4/5) * u

So,

=> $(16/25)u^{2} = u^{2} - 2a(s1)$    

=> $s1 = ( (9/25)*u^{2} ) / 2a$    

For the second plank

=> $0 = (16/25)u^{2} - 2a(s2)$

=> $s2 = ( (16/25)*u^{2} ) / 2a$    

Therefore,

s1 / s1 =  9 / 16

Answer 2

In first, we can find the final velocity (V1) of the bullet after it gone through the first bullet,

V1 = V - V/5

V1 = 4V/5

Then we can find thickness of the first plank(S1) by using a linear motion equation,

$V^{2} = U^{2} + 2aS$----- (1)

Where, V = final velocity of the bullet = 5V/4

            U= initial velocity of the bullet = V

            a = acceleration of the bullet = -a

            S = thickness of the first plank = S1

Substituting above values into the equation (1),

$(4V/5)^{2} = V^{2} - 2aS1\\ 16V^{2}/25 = V^{2} - 2aS1\\  <strong>S1 = 9V^{2} /50a$----- (2)

Then we can find thickness of the second plank(S2) by using equation (1),

            V = 0

            U= 4V/5

            a = -a (∵average resistance is same in both. mass is also equal hence acceleration is equal)

            S = S2

$0 = (4V/5)^{2} - 2aS2\\<strong>S2 = 16V^{2} /50a------- (3)$

Ratio of the thickness of planks=thickness of plank 1/ thickness of plank 2

dividing equation (2) by (3),

Ratio of the thickness of planks = S1/S2 = 9/16