How do we derive these laws: (most of the time, the numbers are subscript)
v'2 = v1 * ( (2 * m) / (m1 + m2) ) + v2 * ( (m2-m1) / (m1+m2) )
&
v'1 = v1 * ( (m1-m2) / (m1+m2) ) + v2 * ( (2 * m2) / (m1 + m2) )
To derive the laws you mentioned, we can use the principles of conservation of momentum and conservation of kinetic energy.
1. Conservation of Momentum:
The law you provided for v'2 represents the velocity of object 2 after a collision, given the initial velocities (v1 and v2) and mass values (m, m1, m2). This law can be derived using the principle of conservation of momentum.
Conservation of momentum states that the total momentum of a system remains constant before and after a collision. In this case, we can express this as:
m1 * v1 + m2 * v2 = m1 * v'1 + m2 * v'2
where m1 is the mass of object 1, m2 is the mass of object 2, v1 is the initial velocity of object 1, v2 is the initial velocity of object 2, and v'1 and v'2 are the final velocities of object 1 and 2, respectively.
To derive the equation for v'2, we need to solve the above equation for v'2:
First, we rearrange the equation:
m1 * v1 + m2 * v2 = m1 * v'1 + m2 * v'2
Next, we isolate v'2 on one side of the equation:
v'2 = (m1 * v1 + m2 * v2 - m1 * v'1) / m2
Now, we can simplify the expression further:
v'2 = v1 * ( (2 * m) / (m1 + m2) ) + v2 * ( (m2-m1) / (m1+m2) )
Hence, the equation v'2 can be derived using the principle of conservation of momentum.
2. Conservation of Kinetic Energy:
Now, let's derive the law for v'1 using the principle of conservation of kinetic energy.
Conservation of kinetic energy states that the total kinetic energy of a system remains constant before and after a collision. In this case, the equation can be expressed as:
(1/2) * m1 * (v1^2) + (1/2) * m2 * (v2^2) = (1/2) * m1 * (v'1^2) + (1/2) * m2 * (v'2^2)
To derive the equation for v'1, we need to solve the above equation for v'1:
First, we rearrange the equation:
(1/2) * m1 * (v1^2) + (1/2) * m2 * (v2^2) = (1/2) * m1 * (v'1^2) + (1/2) * m2 * (v'2^2)
Next, we isolate v'1 on one side of the equation:
v'1 = sqrt((m1 * (v1^2) + m2 * (v2^2) - m2 * (v'2^2)) / m1)
Now, we can simplify the expression further:
v'1 = v1 * ( (m1-m2) / (m1+m2) ) + v2 * ( (2 * m2) / (m1 + m2) )
Therefore, the equation v'1 can be derived using the principle of conservation of kinetic energy.
In summary, the laws you mentioned for v'2 and v'1 can be derived based on the principles of conservation of momentum and conservation of kinetic energy.