To solve the problem of the elastic collision between blocks A and B, we can follow these steps:
### Step 1: Identify the given values
- Mass of block A, \( m_A = 2.0 \, \text{kg} \)
- Velocity of block A, \( v_A = 10 \, \text{m/s} \) (positive x direction)
- Mass of block B, \( m_B = 3.0 \, \text{kg} \)
- Velocity of block B, \( v_B = -5 \, \text{m/s} \) (negative x direction)
### Step 2: Use the conservation of momentum
In an elastic collision, the total momentum before the collision is equal to the total momentum after the collision. The equation for momentum conservation can be written as:
\[
m_A v_A + m_B v_B = m_A v_A' + m_B v_B'
\]
Where \( v_A' \) and \( v_B' \) are the velocities of blocks A and B after the collision.
Substituting the known values:
\[
(2.0 \, \text{kg})(10 \, \text{m/s}) + (3.0 \, \text{kg})(-5 \, \text{m/s}) = (2.0 \, \text{kg})v_A' + (3.0 \, \text{kg})v_B'
\]
Calculating the left side:
\[
20 - 15 = 2v_A' + 3v_B'
\]
This simplifies to:
\[
5 = 2v_A' + 3v_B' \quad \text{(Equation 1)}
\]
### Step 3: Use the elastic collision condition
In an elastic collision, the relative velocity of approach is equal to the relative velocity of separation. This can be expressed as:
\[
v_B - v_A = -(v_B' - v_A')
\]
Substituting the known velocities:
\[
-5 - 10 = -(v_B' - v_A')
\]
This simplifies to:
\[
-15 = -v_B' + v_A' \quad \Rightarrow \quad v_A' - v_B' = 15 \quad \text{(Equation 2)}
\]
### Step 4: Solve the system of equations
Now we have two equations to solve:
1. \( 2v_A' + 3v_B' = 5 \)
2. \( v_A' - v_B' = 15 \)
From Equation 2, we can express \( v_A' \) in terms of \( v_B' \):
\[
v_A' = v_B' + 15
\]
Substituting this into Equation 1:
\[
2(v_B' + 15) + 3v_B' = 5
\]
Expanding and simplifying:
\[
2v_B' + 30 + 3v_B' = 5
\]
\[
5v_B' + 30 = 5
\]
\[
5v_B' = 5 - 30
\]
\[
5v_B' = -25 \quad \Rightarrow \quad v_B' = -5 \, \text{m/s}
\]
Now substituting \( v_B' \) back into Equation 2 to find \( v_A' \):
\[
v_A' = -5 + 15 = 10 \, \text{m/s}
\]
### Final Result
The final velocities after the collision are:
- \( v_A' = 10 \, \text{m/s} \)
- \( v_B' = -5 \, \text{m/s} \)
