Derive the stationary target equations:
v1 = [(m1-m2)/(m1+m2)]v1
v2 = [(2m1)/(m1 + m2)] v1
To derive the stationary target equations, let's start with the conservation of linear momentum. In an isolated system, the total momentum before an event is equal to the total momentum after the event.
Consider a scenario where two objects, object 1 of mass m1 and object 2 of mass m2, collide and stick together. Let v1 be the initial velocity of object 1 before the collision, and v2 be the initial velocity of object 2 before the collision.
Before the collision:
Linear momentum of object 1 = m1 * v1
Linear momentum of object 2 = m2 * v2
After the collision, the two objects stick together:
Mass of the combined system = m1 + m2
Let V be the final velocity of the combined system after the collision.
After the collision:
Linear momentum of the combined system = (m1 + m2) * V
Since momentum is conserved, we can equate the total momentum before and after the collision:
m1 * v1 + m2 * v2 = (m1 + m2) * V
Now we can derive the equations for v1 and v2 in terms of V:
First, solve the equation for V:
(m1 * v1 + m2 * v2) = (m1 + m2) * V
V = (m1 * v1 + m2 * v2) / (m1 + m2)
Now, let's calculate v1:
To find v1, we need to find the final velocity V.
v1 = V
Substitute the calculated value of V:
v1 = (m1 * v1 + m2 * v2) / (m1 + m2)
Now, multiply both sides of the equation by (m1 + m2):
(v1 * (m1 + m2)) = (m1 * v1 + m2 * v2)
Distribute v1:
v1 * m1 + v1 * m2 = m1 * v1 + m2 * v2
Rearrange the terms:
v1 * m1 - m1 * v1 = m2 * v2 - v1 * m2
Factor out m1 on the left side and m2 on the right side:
m1 * (v1 - v1) = m2 * (v2 - v1)
(v1 - v1) is equal to 0, so it simplifies to:
0 = m2 * (v2 - v1)
Divide both sides by m1 + m2:
0 / (m1 + m2) = (m2 * (v2 - v1)) / (m1 + m2)
Simplify to get the final equation for v1:
0 = v2 - v1
This implies that v2 = v1.
Now, let's calculate v2:
Substitute v2 = v1 into the momentum conservation equation:
m1 * v1 + m2 * v2 = (m1 + m2) * V
m1 * v1 + m2 * v1 = (m1 + m2) * V
Combine like terms on the left side:
(v1 * (m1 + m2)) = (m1 + m2) * V
Divide both sides by (m1 + m2):
v1 = V
Substitute the expression for V:
v1 = (m1 * v1 + m2 * v1) / (m1 + m2)
Multiply both sides by (m1 + m2):
v1 * (m1 + m2) = m1 * v1 + m2 * v1
Distribute v1:
v1 * m1 + v1 * m2 = m1 * v1 + m2 * v1
Rearrange the terms:
v1 * m1 - m1 * v1 = m2 * v1 - v1 * m2
Factor out m1:
m1 * (v1 - v1) = v1 * (m2 - m1)
(v1 - v1) is equal to 0, so it simplifies to:
0 = v1 * (m2 - m1)
Divide both sides by (m1 + m2):
0 / (m1 + m2) = v1 * (m2 - m1) / (m1 + m2)
Simplify to get the final equation for v2:
0 = v1
Thus, the final derived stationary target equations are:
v1 = 0
v2 = v1