Two identical blocks of ice, moving with equal velocity v in opposite directions, collide, stick to each other and stop. Find the minimum value of v so that the blocks completely melt aftercollision. Assume that all the heat generated is completely absorbedby the ice blocks. Given: Heat required to convert a kilogram of ice at 0? to water at the same temperature is equal to 3.3 × 100000 J.

To solve this problem, we need to analyze the collision of the two ice blocks and the energy conversion that occurs during and after the collision. The key here is to determine the minimum velocity \( v \) required for the blocks to generate enough heat to completely melt themselves after they collide and stick together.

Understanding the Collision and Energy Transfer

When the two blocks of ice collide, they come to a stop due to their equal and opposite momentum. The kinetic energy that they had before the collision is converted into thermal energy, which can then be used to melt the ice. The total kinetic energy of the system before the collision can be calculated using the formula:

• Kinetic Energy (KE) = \( \frac{1}{2} m v^2 \)

Since there are two identical blocks of ice, each with mass \( m \), the total kinetic energy before the collision is:

• Total KE = \( 2 \times \frac{1}{2} m v^2 = m v^2 \)

Heat Required to Melt the Ice

Next, we need to determine how much heat is required to completely melt the two blocks of ice. The heat required to convert 1 kg of ice at 0°C to water at the same temperature is given as \( 3.3 \times 10^5 \) J. Therefore, for two blocks of ice, the total heat required \( Q \) is:

• Total Heat Required \( Q = 2 \times 3.3 \times 10^5 \, \text{J} = 6.6 \times 10^5 \, \text{J} \)

Setting Up the Energy Balance

For the blocks to completely melt, the kinetic energy generated during the collision must be equal to or greater than the heat required to melt the ice. Thus, we can set up the following equation:

• \( m v^2 \geq 6.6 \times 10^5 \, \text{J} \)

Solving for Minimum Velocity

To find the minimum value of \( v \), we can rearrange the inequality:

• \( v^2 \geq \frac{6.6 \times 10^5 \, \text{J}}{m} \)

Taking the square root of both sides gives us:

• \( v \geq \sqrt{\frac{6.6 \times 10^5}{m}} \)

Final Considerations

At this point, we need to know the mass \( m \) of the ice blocks to calculate the exact minimum velocity. However, the formula \( v \geq \sqrt{\frac{6.6 \times 10^5}{m}} \) provides a clear relationship between the mass of the ice and the minimum velocity required for them to melt completely after the collision. If you have a specific mass for the blocks, you can substitute it into this equation to find the minimum velocity.

In summary, the kinetic energy generated during the collision must be sufficient to provide the heat needed to melt the ice, and this relationship allows us to calculate the minimum velocity required based on the mass of the ice blocks.

To solve the problem of two identical blocks of ice colliding and melting after coming to a stop, we need to apply the principles of conservation of momentum and energy. Let's break this down step by step.

Understanding the Collision

When the two blocks of ice collide, they stick together and come to a stop. The initial kinetic energy of the blocks is transformed into heat energy due to the collision. This heat energy is what will cause the ice to melt.

Conservation of Momentum

Since the blocks are identical and moving in opposite directions with equal velocity, the total momentum before the collision is:

• Momentum of Block 1: \( m \cdot v \)
• Momentum of Block 2: \( -m \cdot v \)

The total momentum before the collision is:

Total Momentum = mv + (-mv) = 0

After the collision, since they stick together and stop, the total momentum is also 0, which is consistent with the conservation of momentum.

Kinetic Energy Before Collision

The kinetic energy (KE) of each block before the collision is given by:

KE = \frac{1}{2} mv^2

Since there are two blocks, the total kinetic energy before the collision is:

Total KE = 2 \cdot \frac{1}{2} mv^2 = mv^2

Heat Required to Melt the Ice

To find the minimum velocity \( v \) required for the blocks to completely melt after the collision, we need to calculate the heat energy required to melt the ice. The heat required to convert 1 kg of ice at 0°C to water at the same temperature is given as:

Q = 3.3 \times 10^5 \, \text{J/kg}

Assuming each block has a mass \( m \), the total heat required to melt both blocks is:

Total Heat Required = 2 \cdot Q = 2 \cdot 3.3 \times 10^5 \, \text{J} = 6.6 \times 10^5 \, \text{J}

Setting Kinetic Energy Equal to Heat Energy

For the blocks to completely melt, the kinetic energy converted into heat must equal the heat required to melt the ice:

mv^2 = 6.6 \times 10^5

Solving for Velocity

Now, we can solve for \( v \):

v^2 = \frac{6.6 \times 10^5}{m}

v = \sqrt{\frac{6.6 \times 10^5}{m}}

Conclusion

The minimum value of \( v \) depends on the mass \( m \) of each block of ice. If you know the mass of the ice blocks, you can substitute that value into the equation to find the minimum velocity required for the blocks to completely melt after the collision. This approach illustrates the interplay between kinetic energy and thermal energy in a collision scenario.