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To solve the projectile motion problem, we need to find the height of the projectile after it has traveled 5 km horizontally.

The key components of the problem are:

- Initial speed (\(v_0 = 450 \, \text{m/s}\))
- Angle of projection (\(\theta = 30^\circ\))
- Horizontal range (\(x = 5000 \, \text{m}\))

We'll break the initial velocity into its horizontal (\(v_{0x}\)) and vertical components (\(v_{0y}\)):
\[
v_{0x} = v_0 \cdot \cos(\theta) = 450 \cdot \cos(30^\circ)
\]
\[
v_{0y} = v_0 \cdot \sin(\theta) = 450 \cdot \sin(30^\circ)
\]

Next, we find the time it takes to travel 5 km horizontally. The horizontal distance traveled is given by:
\[
x = v_{0x} \cdot t
\]
Solving for \(t\):
\[
t = \frac{x}{v_{0x}} = \frac{5000}{450 \cdot \cos(30^\circ)}
\]

Finally, the height at time \(t\) is given by the vertical position equation:
\[
y = v_{0y} \cdot t - \frac{1}{2} g t^2
\]
where \(g = 9.8 \, \text{m/s}^2\) is the acceleration due to gravity.

Let me calculate this step by step.The height of the projectile when it is 5 km downrange is approximately **2080 meters**.

Would you like more details or further clarification?

Here are 5 related questions to consider:
1. How would the height change if the angle of projection were increased?
2. What is the total time of flight for this projectile?
3. What is the maximum height reached by the projectile during its flight?
4. How would the initial speed affect the range of the projectile?
5. How do air resistance and other factors impact the motion of real-world projectiles?

**Tip:** Breaking velocities into horizontal and vertical components is crucial when solving projectile motion problems!

A projectile is fired with an initial speed of 450 m/sec at an angle of elevation of 30 degrees. How high overhead will the projectile be when it is 5 km downrange?

Solve a projectile motion problem where a projectile is fired with a speed of 450 m/s at a 30° angle. Calculate the height of the projectile when it is 5 km downrange, including key concepts like trigonometry and kinematics. Suitable for high school physics students.

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