An explosion breaks an object initially at rest into two pieces, one of which has 2.0 times the mass of the other. If 7300 J of kinetic energy were released in the explosion, how much kinetic energy did the lighter piece acquire?
To find the kinetic energy acquired by the lighter piece, we can use the principle of conservation of energy. According to this principle, the total amount of energy before and after the explosion remains constant.
Let's assume the initial mass of the object is M. Therefore, the mass of the lighter piece would be m, and the other piece would have a mass of 2m.
The kinetic energy of an object is given by the equation:
KE = (1/2) * mass * velocity^2
Since the object was initially at rest, the initial kinetic energy is zero. After the explosion, the total kinetic energy released is 7300 J.
Let's denote the kinetic energy acquired by the lighter piece as KE1 and the kinetic energy acquired by the heavier piece as KE2.
According to the conservation of energy:
KE1 + KE2 = Total kinetic energy released
Substituting the given values:
KE1 + KE2 = 7300 J
Next, we can use the fact that the mass of the lighter piece is 2.0 times smaller than that of the heavier piece:
m = (1/2) * M
2m = M
Now, we can express the kinetic energy of each piece in terms of their masses:
KE1 = (1/2) * m * v1^2
KE2 = (1/2) * (2m) * v2^2 = (1/2) * 2 * m * v2^2 = m * v2^2
We need to find KE1, so we can rearrange the conservation of energy equation to solve for KE1:
KE1 = Total kinetic energy released - KE2
Substituting the forms of KE1 and KE2:
(1/2) * m * v1^2 = 7300 J - m * v2^2
Since we know that m = (1/2) * M, we can substitute this into the equation as well:
(1/2) * (1/2) * M * v1^2 = 7300 J - (1/2) * M * v2^2
Simplifying:
(1/4) * M * v1^2 = 7300 J - (1/2) * M * v2^2
Now, we have two equations involving the masses and velocities of the two pieces. However, we need additional information to determine specific values and solve for KE1.
If you have any additional information, such as the velocities of the pieces or the mass of the entire object (M), please provide them so that we can calculate the kinetic energy acquired by the lighter piece (KE1).