Answer :
To determine how much kinetic energy is lost as a result of the collision between two carts, we need to follow a series of steps.
### Step 1: Determine the Total Mass After Collision
First, let’s calculate the total mass of the two carts after they collide and stick together:
[tex]\[ m_{\text{total}} = m_1 + m_2 \][/tex]
Given:
[tex]\[ m_1 = 0.32 \, \text{kg} \][/tex]
[tex]\[ m_2 = 0.44 \, \text{kg} \][/tex]
Adding these masses together:
[tex]\[ m_{\text{total}} = 0.32 \, \text{kg} + 0.44 \, \text{kg} = 0.76 \, \text{kg} \][/tex]
### Step 2: Calculate the Final Velocity Using Conservation of Momentum
The principle of conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision. Since the second cart is initially at rest, its initial momentum is zero. Thus, the momentum before the collision is just due to the first cart:
[tex]\[ p_{\text{initial}} = m_1 v_0 \][/tex]
After the collision, both carts stick together and move with a common final velocity [tex]\( v_{\text{final}} \)[/tex]. So, the total momentum after the collision is:
[tex]\[ p_{\text{final}} = m_{\text{total}} v_{\text{final}} \][/tex]
Setting the initial momentum equal to the final momentum:
[tex]\[ m_1 v_0 = m_{\text{total}} v_{\text{final}} \][/tex]
Solving for [tex]\( v_{\text{final}} \)[/tex]:
[tex]\[ v_{\text{final}} = \frac{m_1 v_0}{m_{\text{total}}} \][/tex]
Substituting the known values:
[tex]\[ m_1 = 0.32 \, \text{kg} \][/tex]
[tex]\[ v_0 = 0.85 \, \text{m/s} \][/tex]
[tex]\[ m_{\text{total}} = 0.76 \, \text{kg} \][/tex]
[tex]\[ v_{\text{final}} = \frac{0.32 \, \text{kg} \times 0.85 \, \text{m/s}}{0.76 \, \text{kg}} \approx 0.3579 \, \text{m/s} \][/tex]
### Step 3: Calculate the Initial Kinetic Energy
The initial kinetic energy is only due to the first cart, since the second cart is initially at rest. The kinetic energy (KE) is given by:
[tex]\[ KE_{\text{initial}} = \frac{1}{2} m_1 v_0^2 \][/tex]
Substituting the known values:
[tex]\[ KE_{\text{initial}} = \frac{1}{2} \times 0.32 \, \text{kg} \times (0.85 \, \text{m/s})^2 \approx 0.1156 \, \text{J} \][/tex]
### Step 4: Calculate the Final Kinetic Energy
After the collision, both carts move together with a common velocity [tex]\( v_{\text{final}} \)[/tex]. Thus, the final kinetic energy is:
[tex]\[ KE_{\text{final}} = \frac{1}{2} m_{\text{total}} v_{\text{final}}^2 \][/tex]
Using the known values:
[tex]\[ m_{\text{total}} = 0.76 \, \text{kg} \][/tex]
[tex]\[ v_{\text{final}} = 0.3579 \, \text{m/s} \][/tex]
[tex]\[ KE_{\text{final}} = \frac{1}{2} \times 0.76 \, \text{kg} \times (0.3579 \, \text{m/s})^2 \approx 0.04867 \, \text{J} \][/tex]
### Step 5: Calculate the Kinetic Energy Lost
The kinetic energy lost is the difference between the initial and final kinetic energies:
[tex]\[ KE_{\text{lost}} = KE_{\text{initial}} - KE_{\text{final}} \][/tex]
Using the calculated values:
[tex]\[ KE_{\text{initial}} \approx 0.1156 \, \text{J} \][/tex]
[tex]\[ KE_{\text{final}} \approx 0.04867 \, \text{J} \][/tex]
[tex]\[ KE_{\text{lost}} \approx 0.1156 \, \text{J} - 0.04867 \, \text{J} \approx 0.06693 \, \text{J} \][/tex]
### Conclusion
The kinetic energy lost as a result of the collision is approximately [tex]\( 0.06693 \)[/tex] joules.
### Step 1: Determine the Total Mass After Collision
First, let’s calculate the total mass of the two carts after they collide and stick together:
[tex]\[ m_{\text{total}} = m_1 + m_2 \][/tex]
Given:
[tex]\[ m_1 = 0.32 \, \text{kg} \][/tex]
[tex]\[ m_2 = 0.44 \, \text{kg} \][/tex]
Adding these masses together:
[tex]\[ m_{\text{total}} = 0.32 \, \text{kg} + 0.44 \, \text{kg} = 0.76 \, \text{kg} \][/tex]
### Step 2: Calculate the Final Velocity Using Conservation of Momentum
The principle of conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision. Since the second cart is initially at rest, its initial momentum is zero. Thus, the momentum before the collision is just due to the first cart:
[tex]\[ p_{\text{initial}} = m_1 v_0 \][/tex]
After the collision, both carts stick together and move with a common final velocity [tex]\( v_{\text{final}} \)[/tex]. So, the total momentum after the collision is:
[tex]\[ p_{\text{final}} = m_{\text{total}} v_{\text{final}} \][/tex]
Setting the initial momentum equal to the final momentum:
[tex]\[ m_1 v_0 = m_{\text{total}} v_{\text{final}} \][/tex]
Solving for [tex]\( v_{\text{final}} \)[/tex]:
[tex]\[ v_{\text{final}} = \frac{m_1 v_0}{m_{\text{total}}} \][/tex]
Substituting the known values:
[tex]\[ m_1 = 0.32 \, \text{kg} \][/tex]
[tex]\[ v_0 = 0.85 \, \text{m/s} \][/tex]
[tex]\[ m_{\text{total}} = 0.76 \, \text{kg} \][/tex]
[tex]\[ v_{\text{final}} = \frac{0.32 \, \text{kg} \times 0.85 \, \text{m/s}}{0.76 \, \text{kg}} \approx 0.3579 \, \text{m/s} \][/tex]
### Step 3: Calculate the Initial Kinetic Energy
The initial kinetic energy is only due to the first cart, since the second cart is initially at rest. The kinetic energy (KE) is given by:
[tex]\[ KE_{\text{initial}} = \frac{1}{2} m_1 v_0^2 \][/tex]
Substituting the known values:
[tex]\[ KE_{\text{initial}} = \frac{1}{2} \times 0.32 \, \text{kg} \times (0.85 \, \text{m/s})^2 \approx 0.1156 \, \text{J} \][/tex]
### Step 4: Calculate the Final Kinetic Energy
After the collision, both carts move together with a common velocity [tex]\( v_{\text{final}} \)[/tex]. Thus, the final kinetic energy is:
[tex]\[ KE_{\text{final}} = \frac{1}{2} m_{\text{total}} v_{\text{final}}^2 \][/tex]
Using the known values:
[tex]\[ m_{\text{total}} = 0.76 \, \text{kg} \][/tex]
[tex]\[ v_{\text{final}} = 0.3579 \, \text{m/s} \][/tex]
[tex]\[ KE_{\text{final}} = \frac{1}{2} \times 0.76 \, \text{kg} \times (0.3579 \, \text{m/s})^2 \approx 0.04867 \, \text{J} \][/tex]
### Step 5: Calculate the Kinetic Energy Lost
The kinetic energy lost is the difference between the initial and final kinetic energies:
[tex]\[ KE_{\text{lost}} = KE_{\text{initial}} - KE_{\text{final}} \][/tex]
Using the calculated values:
[tex]\[ KE_{\text{initial}} \approx 0.1156 \, \text{J} \][/tex]
[tex]\[ KE_{\text{final}} \approx 0.04867 \, \text{J} \][/tex]
[tex]\[ KE_{\text{lost}} \approx 0.1156 \, \text{J} - 0.04867 \, \text{J} \approx 0.06693 \, \text{J} \][/tex]
### Conclusion
The kinetic energy lost as a result of the collision is approximately [tex]\( 0.06693 \)[/tex] joules.
