A 8.00 kg shell at rest explodes into two fragments, one with a mass of 2.00 kg and the other with a mass of 6.00 kg.
If the heavier fragment gains 130 J of kinetic energy from the explosion, how much kinetic energy does the lighter one gain?
You have to find the velocity of the heavier fragment, then its momentum. The lighter fragment will have opposite momentum, from which you can find its velocity.
2 piece shell have same kinetic energy u can find one piece velocity by apply 1/2m(v)2 rule
To determine the kinetic energy gained by the lighter fragment, we need to understand the principle of conservation of momentum. According to this principle, the total momentum before the explosion is equal to the total momentum after the explosion.
Before the explosion, the shell is at rest, so its momentum is zero.
After the explosion, the total momentum of the two fragments must also be zero since there is no external force acting on them. We can represent the momentum of the heavier fragment as p1 and the momentum of the lighter fragment as p2.
So, p1 + p2 = 0 ---- (Equation 1)
We can find the momentum of an object using the formula:
Momentum (p) = mass (m) × velocity (v)
Since the shell is at rest before the explosion, the velocity of both fragments is unknown. However, we can assume that the explosion occurs in an isolated system, which means no external forces act on it. Hence, the momentum of the system is always conserved.
Using this information, we have:
p1 = (mass of heavier fragment) × (velocity of heavier fragment)
p1 = 6.00 kg × v1 ---- (Equation 2)
p2 = (mass of lighter fragment) × (velocity of lighter fragment)
p2 = 2.00 kg × v2 ---- (Equation 3)
Substituting equations 2 and 3 into equation 1, we get:
6.00 kg × v1 + 2.00 kg × v2 = 0 ---- (Equation 4)
Now, since the heavier fragment gains 130 J of kinetic energy, we can relate kinetic energy (K) to momentum using the formula:
Kinetic Energy (K) = (1/2) × mass × velocity^2
Applying this formula to the heavier fragment, we have:
130 J = (1/2) × 6.00 kg × v1^2
Simplifying this equation, we find:
v1^2 = (2 × 130 J) / 6.00 kg
v1^2 = 260 J / 6.00 kg
v1^2 = 43.333 J/kg
Now, since the kinetic energy gained by an object is directly proportional to the square of its velocity, we can use this information to find the kinetic energy gained by the lighter fragment.
The ratio of the velocities of the two fragments can be calculated using equation 4:
v2 = - (6.00 kg × v1) / 2.00 kg
v2 = - (6.00 / 2.00) × v1
v2 = - 3.00 × v1
Substituting the value of v1^2, we have:
v2 = - 3.00 × √(43.333 J/kg)
Now, finding the kinetic energy of the lighter fragment using the formula:
Kinetic Energy (K) = (1/2) × mass × velocity^2
K = (1/2) × 2.00 kg × (- 3.00 × √(43.333 J/kg))^2
K = (1/2) × 2.00 kg × 3.00^2 × 43.333 J
K = 1169.99 J
Therefore, the lighter fragment gains approximately 1169.99 J of kinetic energy from the explosion.