An explosion breaks a 10 kg object (originally at rest) into two pieces, one of which is two times the mass of the
other. If 8000 J is released in the explosion, a) how fast are the pieces moving away from their original location?
b)how much kinetic energy does each piece have after the explosion?
c) Discuss whether or not your answers to parts a and b are reasonable. Include specific reasons why you think
they are or are not reasonable.
a) To find the velocity at which the pieces are moving away from their original location, we can use 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.
Initially, the object is at rest, so its momentum is zero. The total mass of the object is 10 kg, and it breaks into two pieces, one with a mass of 2x kg and the other with a mass of x kg.
Let's assume the velocity of the lighter piece (mass x) is v1 and the velocity of the heavier piece (mass 2x) is v2.
The total momentum before the explosion is 0, and after the explosion, it remains 0. Therefore, we can write the conservation of momentum equation as:
0 = (2x)(v2) + (x)(v1)
Since the lighter piece and the heavier piece move in opposite directions, we can assign opposite signs to their velocities. Let's say the lighter piece moves to the right and the heavier piece moves to the left. Then, v1 = -v2.
0 = (2x)(-v2) + (x)(v1)
0 = -2xv2 + xv1
0 = -v2 + 0.5v1
Simplifying the equation, we get:
v2 = 0.5v1
Now we can use the fact that 8000 J is released in the explosion to find the velocities. The kinetic energy released in the explosion is equal to the sum of the kinetic energies of the two pieces.
b) The kinetic energy of an object is given by the equation KE = (1/2)mv^2, where KE is the kinetic energy, m is the mass, and v is the velocity.
Let's calculate the kinetic energy of each piece after the explosion:
For the lighter piece:
KE1 = (1/2)(x)(v1^2)
For the heavier piece:
KE2 = (1/2)(2x)(v2^2)
Summing up the kinetic energies:
KE_total = KE1 + KE2
Now let's plug in the values: KE_total = (1/2)(x)(v1^2) + (1/2)(2x)(v2^2)
c) To determine whether our answers are reasonable, let's analyze the situation.
Since the initial object was at rest, it is logical to assume that the total momentum after the explosion should be zero. Our initial assumption was correct, as shown by the conservation of momentum equation. This indicates that our answer to part a) is reasonable.
In part b), we calculated the kinetic energy of each piece after the explosion. The sum of these kinetic energies should be equal to the total kinetic energy released in the explosion. It is logical that the total kinetic energy released should be equal to the energy released in the explosion. If our answer to part b) satisfies this condition, then it is reasonable.
To check the reasonableness of our answer to part b), we can compare the total kinetic energy calculated with the given energy of 8000 J. If the sum of the kinetic energies equals 8000 J, then our answer is reasonable. However, if it does not match, we would need to reassess our calculations and find any potential errors.
By analyzing the situation and comparing our answers to the conditions and given information, we can determine the reasonableness of our solutions.
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