An explosion breaks an object into two pieces, one of which has 3.20 times the mass of the other. If 8000 J were released in the explosion, how much kinetic energy did each piece acquire?
I started with using KE = (1/2)mv^2 and m1v1 = m2v2. Then (1/2)m1(v1)^2 + (1/2)m2(v2)^2 = 8000J. How do I find the velocities? I tried rearranging the equations to solve for one of the unknowns, but got stuck.
Momentum is conserved. The initial momentum is zero, therefore:
m1 v1 + m2 v2 = 0 ---->
m1 v1 = m2 v2 ----->
(m1 v1)^2 = (m2 v2)^2
1/2 m1 v1^2 = 1/2 m2 v2^2 * (m2/m1)
So, you have:
KE1 + KE2 = 8000 J
and:
KE1 = (m2/m1) KE2
Well, we know that KE1 + KE2 = 8000 J, and KE1 = (m2/m1) KE2. Let's substitute KE1 in the first equation with this expression:
(m2/m1) KE2 + KE2 = 8000 J
Now, let's combine like terms:
[(m2/m1) + 1] KE2 = 8000 J
To find the value of KE2, we can divide both sides of the equation by [(m2/m1) + 1]:
KE2 = 8000 J / [(m2/m1) + 1]
Now we can substitute this value of KE2 back into the equation for KE1:
KE1 = (m2/m1) KE2
KE1 = (m2/m1) * (8000 J / [(m2/m1) + 1])
Now we have expressions for both KE1 and KE2. All that's left is to calculate their values!
To find the velocities, we can use the fact that kinetic energy is given by KE = (1/2)mv^2. We already have the relationship m1v1 = m2v2 from conservation of momentum.
Let's substitute this relationship into the equation KE1 = (m2/m1) KE2:
(1/2)m1v1^2 = (m2/m1)(1/2)m2v2^2
Now, let's simplify the equation:
m1v1^2 = m2^2v2^2/m1
Multiply both sides by m1 to get rid of the fraction:
m1^2v1^2 = m2^2v2^2
Taking the square root of both sides, we obtain:
m1v1 = m2v2
Now, we have two simultaneous equations:
m1v1 + m2v2 = 0 (from conservation of momentum)
m1v1 = m2v2
To solve for the velocities, we can use the second equation to substitute for v1 in the first equation:
m2v2 + m2v2 = 0
2m2v2 = 0
v2 = 0
Substituting this value into the equation m1v1 = m2v2, we get:
m1v1 = 0
Therefore, the velocities are v1 = 0 and v2 = 0.
Since both velocities are zero, each piece acquired no kinetic energy from the explosion.
To find the velocities, let's start by substituting the expression for KE1 in terms of KE2:
(m2/m1) KE2 + KE2 = 8000 J
Now, let's combine the terms:
[(m2/m1) + 1] KE2 = 8000 J
To isolate KE2, divide both sides by [(m2/m1) + 1]:
KE2 = 8000 J / [(m2/m1) + 1]
Now that we have the value for KE2, we can substitute it back into the equation for KE1 to obtain its value:
KE1 = (m2/m1) KE2
Now, let's plug in the values given:
m2 = 3.20m1
KE1 = 3.20 KE2
Now substitute the expression for KE2 in terms of m1 and m2:
KE1 = 3.20 * (8000 J / [(m2/m1) + 1])
Simplifying further:
KE1 = 3.20 * (8000 J / [(3.20m1/m1) + 1])
KE1 = 3.20 * (8000 J / [3.20 + 1])
KE1 = 3.20 * (8000 J / 4.20)
KE1 = 6153.85 J
Therefore, each piece acquires 6153.85 J of kinetic energy.