Suppose someone is standing on top of a cliff and launches a rock upward at an initial speed of 40 m/s and at an initial height of 50 meters above the ground at the base of the cliff.  

Determine the maximum height of the rock, in meters, above the ground. 
 

To determine the maximum height of the rock, we need to calculate two key aspects of its motion: the point where its velocity becomes zero (the maximum height), and then add that height to the initial height above the ground.

### Step 1: Use the kinematic equation to find the maximum height above the launch point.

The kinematic equation for vertical motion is:

\[
v^2 = u^2 + 2a s
\]

Where:
- \( v \) is the final velocity at the highest point (0 m/s, since the rock momentarily stops).
- \( u \) is the initial velocity (40 m/s).
- \( a \) is the acceleration (gravity, which is \(-9.8 \, \text{m/s}^2\)).
- \( s \) is the displacement from the initial position to the highest point.

We need to solve for \( s \) (the height above the launch point).

### Step 2: Plug in the values.

\[
0 = (40)^2 + 2(-9.8) s
\]

\[
0 = 1600 - 19.6 s
\]

\[
19.6 s = 1600
\]

\[
s = \frac{1600}{19.6} \approx 81.63 \, \text{meters}
\]

### Step 3: Calculate the total maximum height.

The total height above the ground is the sum of the initial height (50 m) and the displacement from the launch point to the highest point:

\[
\text{Total maximum height} = 50 \, \text{m} + 81.63 \, \text{m} = 131.63 \, \text{meters}
\]

Thus, the maximum height of the rock above the ground is approximately **131.63 meters**.

---

Would you like a detailed breakdown of any part of this process? 

Here are 5 questions that expand on the topic:
1. How does air resistance affect the maximum height in a real-world scenario?
2. What if the initial velocity was halved? How would the height change?
3. Can you calculate the time it takes to reach the maximum height?
4. How would you determine the total time of flight before the rock hits the ground?
5. What would be the rock's velocity just before it hits the ground?

**Tip:** Always check if the reference point (height) is from the launch point or from the ground when solving kinematic problems.

Suppose someone is standing on top of a cliff and launches a rock upward at an initial speed of 40 m/s and at an initial height of 50 meters above the ground at the base of the cliff. Determine the maximum height of the rock, in meters, above the ground.

Calculate the maximum height of a rock launched from a cliff at an initial speed of 40 m/s and 50 meters above the ground. This solution explains key kinematic concepts and uses vertical motion equations to find the height, ideal for high school physics students.

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