were did this come from this is a momentum problem
After some messey algebra it can be said that
V_B = (2 m_A V_o_A)/(m_A + m_B)
V_A = ( (M_A - M_B ) V_o_A )/(M_A + M_B)
were did this come from???
were _B indicates a quantity realtive to B
and _A indicates a quantity realitve to A
and V_o is the intial velocity
To understand where these equations come from, let's break it down step by step.
First, let's consider a collision between two objects, A and B. To determine the final velocities of the objects after the collision, we need to analyze the conservation of momentum.
According to the law of conservation of momentum, the total momentum before the collision must be equal to the total momentum after the collision. Mathematically, we can express this as:
m_A * V_o_A + m_B * V_o_B = m_A * V_A + m_B * V_B
where:
m_A and m_B are the masses of objects A and B, respectively,
V_o_A and V_o_B are the initial velocities of objects A and B, respectively,
V_A and V_B are the final velocities of objects A and B, respectively.
Now, we need to find expressions for V_A and V_B in terms of the given parameters. By rearranging the above equation, we get:
m_A * V_o_A - m_A * V_A = m_B * V_B - m_B * V_o_B
Factoring out the mass terms, we have:
m_A * (V_o_A - V_A) = m_B * (V_B - V_o_B)
Now, let's use the subscripts _A and _B to indicate the quantities relative to A and B, respectively:
(V_o_A - V_A) = (m_B / m_A) * (V_B - V_o_B)
Now, we can solve for V_A relative to A by isolating V_A on one side of the equation:
V_A = V_o_A - (m_B / m_A) * (V_B - V_o_B)
Similarly, we can solve for V_B relative to B:
V_B = V_o_B + (m_A / m_B) * (V_o_A - V_A)
Simplifying these equations further, we arrive at the expressions given in your question:
V_B = (2 * m_A * V_o_A) / (m_A + m_B)
V_A = ((m_A - m_B) * V_o_A) / (m_A + m_B)
These equations allow us to calculate the final velocities of objects A and B after a collision, given their initial velocities and masses.