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calculate greatest height time in air and horizontal range using suvat

Let's break this down and solve it step by step using the projectile motion concepts.

### Given:
- Initial velocity \( u = 21 \, \text{m/s} \) at an angle \( \theta = 15^\circ \) from the horizontal.
- The height of the cliff \( h = 80 \, \text{m} \).
- Acceleration due to gravity \( g = 9.81 \, \text{m/s}^2 \).

We'll calculate:
1. The greatest height (above the cliff).
2. The time in the air.
3. The horizontal range.

### Resolving the velocity:
- The horizontal component of the velocity:
  \[
  u_x = u \cos \theta = 21 \cos(15^\circ)
  \]
- The vertical component of the velocity:
  \[
  u_y = u \sin \theta = 21 \sin(15^\circ)
  \]

Let's calculate these values first.The initial velocity components are:
- Horizontal component: \( u_x = 20.28 \, \text{m/s} \)
- Vertical component: \( u_y = 5.44 \, \text{m/s} \)

Now, let's calculate the required quantities:

### 1. **Greatest height above the cliff:**
The greatest height is reached when the vertical velocity becomes zero. Using the equation:
\[
v_y^2 = u_y^2 - 2g \Delta y
\]
At the maximum height, \( v_y = 0 \), so:
\[
0 = u_y^2 - 2g \Delta y
\]
Solving for \( \Delta y \) (the height above the cliff):
\[
\Delta y = \frac{u_y^2}{2g}
\]

### 2. **Total time in the air:**
The total time includes two parts:
- Time to reach the maximum height.
- Time to fall down to the ground from the maximum height (which includes the height of the cliff).

The time to reach maximum height:
\[
t_{\text{up}} = \frac{u_y}{g}
\]

To find the total time in the air, we use the equation of motion for the entire fall from the highest point down to the ground. The total vertical displacement is \( h + \Delta y \). Using the equation:
\[
y = u_y t + \frac{1}{2} g t^2
\]
for the downward motion, we solve for the total time.

### 3. **Horizontal range:**
The horizontal range can be calculated by:
\[
R = u_x \times t_{\text{total}}
\]

Let’s calculate these values.It seems like I can’t do more advanced data analysis right now. Please try again later. Would you like me to explain the next steps so you can calculate it yourself? 

Here are a few related questions to think about:
1. How would the range change if the angle of projection increased?
2. What is the role of gravity in determining the maximum height?
3. How does the vertical velocity change over time during projectile motion?
4. What would happen if the object was launched with the same speed but from ground level?
5. How does air resistance affect the motion of a projectile?

**Tip**: When solving for projectile motion, always split the motion into horizontal and vertical components. Each behaves independently, with gravity only affecting the vertical motion.

Explore how to calculate the greatest height, time in the air, and horizontal range for a projectile launched from a cliff using initial velocity, angle, and height data. This analysis uses SUVAT equations and basic trigonometry, ideal for high school students.

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