A mysterious crate has shown up at your place of work, Firecracker Company, and you are told to measure its inertia. It is too heavy to lift, but it rolls smoothly on casters. Getting an inspiration, you lightly tape a 0.60-kg iron block to the side of the crate, slide a firecracker between the crate and the block, and light the fuse. When the firecracker explodes, the block goes one way and the crate rolls the other way. You measure the crate's speed to be 0.058 m/s by timing how long it takes to cross floor tiles. You look up the specifications of the firecracker and find that it releases 7 J of energy. That's all you need, and you quickly calculate the inertia of the crate. What is that inertia?

the inertia of the crate is (49.67 kg)r²

Explanation:

Given the data in the question;

First; we will use the law of conservation of momentum to determine the mass of the crate;

m₁v₁ - m₂v₂ = 0

given that; m₁ = 0.60 kg and v₂ = 0.058 m/s

we substitute

0.60 × v₁ = m₂ × 0.058 = 0

m₂ = 0.60v₁ / 0.058 EQU 1

Next, we use the energy conservation relation to find the velocity

According to conservation of energy;

1/2m₁v₁² + 1/2m₂v₂² = 7 J

we substitute

1/2×0.60×v₁² + 1/2×m₂×(0.058)² = 7 J

0.3v₁² + 0.001682m₂ = 7 J EQU 2

substitute value of m₂ form equ 1 into equ 2

0.3v₁² + 0.001682(0.60v₁ / 0.058) = 7 J

0.3v₁² + 0.0174v₁ = 7 J

0.3v₁² + 0.0174v₁ - 7 J = 0

we solve the quadratic equation;

{  x =  [-b±√( b² - 4ac)] / 2a   }

v₁  =  [-0.0174 ±√( 0.0174² - 4×0.3×-7)] / 2×0.3

=  [-0.0174 ±√(8.4003)] / 0.6

= [-0.0174 ± 2.8983 ] / 0.6  

= -4.8595 or 4.8015     but{ v₁ ≠ - }

so v₁ = 4.8015 m/s ≈ 4.802 m/s

next we input value of  v₁ into equation 1

m₂ = (0.60×4.8015) / 0.058

m₂ =  2.8809 / 0.058

m₂ =  49.67 kg

So, the moment of inertia of the crate will be;

I₂ = m₂r²

we substitute value of m₂

I₂ = (49.67 kg)r²

Therefore, the inertia of the crate is (49.67 kg)r²