To calculate the kinetic energy lost in this collision, we need to follow these steps:

Step 1: Calculate the Initial Kinetic Energies

First, we calculate the initial kinetic energy of each object before the collision.

Object 1

• Mass, $m_1 = 1.2 \, \text{kg}$
• Velocity, $v_{1x} = 25.0 \, \text{m/s}$

The kinetic energy for object 1 is: $KE_1 = \frac{1}{2} m_1 v_{1x}^2$ $KE_1 = \frac{1}{2} \cdot 1.2 \cdot (25.0)^2$ $KE_1 = 0.6 \cdot 625 = 375 \, \text{J}$

Object 2

• Mass, $m_2 = 0.65 \, \text{kg}$
• Velocity, $v_{2y} = 35.0 \, \text{m/s}$

The kinetic energy for object 2 is: $KE_2 = \frac{1}{2} m_2 v_{2y}^2$ $KE_2 = \frac{1}{2} \cdot 0.65 \cdot (35.0)^2$ $KE_2 = 0.325 \cdot 1225 = 398.125 \, \text{J}$

Total Initial Kinetic Energy

$KE_{\text{initial}} = KE_1 + KE_2$ $KE_{\text{initial}} = 375 + 398.125 = 773.125 \, \text{J}$

Step 2: Calculate the Final Velocity of the Combined Object

Since the objects stick together, we have an inelastic collision. We use the conservation of momentum to find the final velocity of the combined object.

The combined mass $M$ of the two objects is: $M = m_1 + m_2 = 1.2 + 0.65 = 1.85 \, \text{kg}$

The initial momentum in the $x$- and $y$-directions are: $p_{x} = m_1 v_{1x} = 1.2 \cdot 25.0 = 30.0 \, \text{kg} \cdot \text{m/s}$ $p_{y} = m_2 v_{2y} = 0.65 \cdot 35.0 = 22.75 \, \text{kg} \cdot \text{m/s}$

The final velocity components $v_{fx}$ and $v_{fy}$ of the combined object are: $v_{fx} = \frac{p_{x}}{M} = \frac{30.0}{1.85} = 16.216 \, \text{m/s}$ $v_{fy} = \frac{p_{y}}{M} = \frac{22.75}{1.85} = 12.297 \, \text{m/s}$

Now we calculate the magnitude of the final velocity $v_f$: $v_f = \sqrt{v_{fx}^2 + v_{fy}^2}$ $v_f = \sqrt{(16.216)^2 + (12.297)^2}$ $v_f = \sqrt{262.97 + 151.24} = \sqrt{414.21} = 20.36 \, \text{m/s}$

Step 3: Calculate the Final Kinetic Energy of the Combined Object

The final kinetic energy $KE_{\text{final}}$ of the combined object is: $KE_{\text{final}} = \frac{1}{2} M v_f^2$ $KE_{\text{final}} = \frac{1}{2} \cdot 1.85 \cdot (20.36)^2$ $KE_{\text{final}} = 0.925 \cdot 414.21 = 383.14 \, \text{J}$

Step 4: Calculate the Kinetic Energy Lost

The kinetic energy lost in the collision is the difference between the initial and final kinetic energies: $\Delta KE = KE_{\text{initial}} - KE_{\text{final}}$ $\Delta KE = 773.125 - 383.14 = 389.985 \, \text{J}$

Answer

The kinetic energy lost in the collision is approximately: $\Delta KE \approx 390 \, \text{J}$

Would you like further details or have any questions?

Here are some related questions for practice:

1. How would the kinetic energy lost change if the masses of the objects were different?
2. What would happen to the kinetic energy lost if the objects did not stick together?
3. How does the direction of the initial velocities affect the final kinetic energy?
4. Can you apply the same approach to a collision where objects do not stick together?
5. How does conservation of momentum differ in elastic and inelastic collisions?

Tip: In inelastic collisions, the kinetic energy is not conserved, but momentum always is.