A block of ice with mass 2.00 kg slides 0.750 m down an inclined
plane that slopes downward at an angle of below the horizontal. If the
block of ice starts from rest, what is its final speed? You can ignore
friction.

To find the final speed of the block of ice sliding down the inclined plane, we can use the principle of energy conservation. Since friction is ignored, the total mechanical energy is conserved.

### Step-by-Step Solution:

1. **Identify the given information:**
   - Mass of the ice block, \( m = 2.00 \, \text{kg} \)
   - Distance traveled along the incline, \( d = 0.750 \, \text{m} \)
   - Angle of the incline, \( \theta \)
   - The block starts from rest, so initial velocity \( v_0 = 0 \, \text{m/s} \)
   - Gravitational acceleration, \( g = 9.81 \, \text{m/s}^2 \)

2. **Calculate the height \( h \) the block descends:**
   The height \( h \) is related to the distance traveled down the slope by:
   \[
   h = d \cdot \sin(\theta)
   \]
   where \( d = 0.750 \, \text{m} \) and \( \theta \) is the angle of the incline.

3. **Use conservation of mechanical energy:**
   The initial mechanical energy of the system is all potential energy, and the final mechanical energy is all kinetic energy, as the block starts from rest and there is no friction. The conservation of energy equation is:
   \[
   mgh = \frac{1}{2} m v^2
   \]
   where:
   - \( m \) is the mass of the block,
   - \( g \) is the gravitational acceleration,
   - \( h \) is the vertical height,
   - \( v \) is the final speed of the block.

4. **Solve for the final speed \( v \):**
   Rearranging the equation for \( v \):
   \[
   v = \sqrt{2gh}
   \]
   Substituting \( h = d \cdot \sin(\theta) \):
   \[
   v = \sqrt{2g \cdot d \cdot \sin(\theta)}
   \]
   Now substitute the known values:
   \[
   v = \sqrt{2 \cdot 9.81 \, \text{m/s}^2 \cdot 0.750 \, \text{m} \cdot \sin(\theta)}
   \]

5. **Final expression:**
   The final speed \( v \) depends on the angle \( \theta \). If you provide the angle, we can calculate the exact final speed.

Would you like to provide the angle, or ask further questions? 

### Related Questions:
1. What is the relationship between the angle of incline and the speed of an object sliding down it?
2. How does friction affect the final speed in a similar scenario?
3. If the mass of the block increased, how would that affect the final speed?
4. What if the incline were not smooth and included friction? How would you solve this problem?
5. How is the principle of conservation of energy applied to systems with non-conservative forces?

### Tip:
Always check if friction can be neglected. In real-world scenarios, friction often has a significant effect on the final speed.

A block of ice with mass 2.00 kg slides 0.750 m down an inclined plane that slopes downward at an angle of below the horizontal. If the block of ice starts from rest, what is its final speed? You can ignore friction.

Learn how to calculate the final speed of a 2.00 kg ice block sliding 0.750 m down a frictionless inclined plane using the conservation of energy. Includes step-by-step explanation of key physics concepts and formulas. Suitable for students in Grades 10-12.

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