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Let's break down and solve the problems one by one based on the information in the image:

### Problem 6:
This problem involves a car moving at a constant speed down a hill. It asks about the relationship between changes in kinetic energy (ΔEk) and gravitational potential energy (ΔEp).

- Since the car is moving at constant speed, its **kinetic energy (Ek)** does not change, implying **ΔEk = 0**.
- The car is moving downhill, so its **gravitational potential energy (Ep)** is decreasing. Thus, **ΔEp** is negative.
  
Given this scenario, the correct statement would be:
- **ΔEk = 0** and **ΔEp < 0**.

Therefore, **Option D: ΔEk = 0** is the correct answer.

---

### Problem 7:
This problem involves calculating the minimum power required to pump rainwater to prevent flooding in a low-lying area. The area receives 1.3 × 10⁹ kg of rain per day, and the water needs to be pumped up 2 meters (the average height below sea level) to prevent flooding.

#### Step-by-Step Solution:
1. **Gravitational potential energy (Ep)** required to pump the water:
   \[
   \Delta E_p = m \cdot g \cdot h
   \]
   where:
   - \( m = 1.3 \times 10^9 \, \text{kg} \) (mass of water),
   - \( g = 9.81 \, \text{m/s}^2 \) (acceleration due to gravity),
   - \( h = 2.0 \, \text{m} \) (height water is lifted).

   Substituting the values:
   \[
   \Delta E_p = 1.3 \times 10^9 \cdot 9.81 \cdot 2
   \]
   \[
   \Delta E_p = 2.55 \times 10^{10} \, \text{J}
   \]

2. **Power required**:
   Power is the rate at which work is done, i.e.,
   \[
   P = \frac{E}{t}
   \]
   The time period given is **one day** or \( 86400 \, \text{seconds} \).

   Substituting the energy calculated:
   \[
   P = \frac{2.55 \times 10^{10}}{86400} \, \text{W}
   \]
   \[
   P = 295,138 \, \text{W} \approx 300 \, \text{kW}
   \]

Thus, the minimum pump output power required is approximately **300 kW**, which corresponds to **Option D**.

---

Would you like further details on any of these solutions?

### Related Questions:
1. How does the change in gravitational potential energy affect the car's motion when moving uphill at constant speed?
2. What would happen to the kinetic energy if the car increased its speed while moving downhill?
3. How would you calculate the energy lost due to friction in Problem 6 if the speed varied?
4. What is the relationship between gravitational potential energy and height for different masses?
5. How would the power calculation change in Problem 7 if the height was doubled?

**Tip**: When solving physics problems, clearly identify whether energy is conserved or lost to friction/heat, and whether velocities are constant or changing.

6. A car driver adjusts the pressure on a car's brakes so that the car travels at constant speed down a hill from P to Q. The magnitude of the change in the car's kinetic energy is ΔEk. The magnitude of the change in its gravitational potential energy is ΔEp. Which statement is correct?
7. An area of land is an average of 2.0m below sea level. To prevent flooding, pumps are used to lift rainwater up to sea level. What is the minimum pump output power required to deal with 1.3×10⁹ kg of rain per day?

Learn how to solve physics problems involving energy conservation, specifically calculating the kinetic and gravitational potential energy for a car moving downhill and the power needed to pump rainwater in a flood scenario. Includes step-by-step explanations suitable for high school students (Grades 10-12).

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