A projectile is fired horizontally at an angle of 45° with the speed of 0.075km/s. Find the range and height. The final answer should be in meters.

To solve for the range and maximum height of a projectile fired at an angle, we can use the following equations of motion for projectile motion.

### Given:
- Initial velocity, \( v_0 = 0.075 \, \text{km/s} = 75 \, \text{m/s} \)
- Angle of projection, \( \theta = 45^\circ \)
- Acceleration due to gravity, \( g = 9.8 \, \text{m/s}^2 \)

### 1. Time of Flight (T)
The time of flight \( T \) for a projectile is given by:

\[
T = \frac{2 v_0 \sin \theta}{g}
\]

### 2. Range (R)
The range \( R \) of the projectile is given by:

\[
R = \frac{v_0^2 \sin(2\theta)}{g}
\]

### 3. Maximum Height (H)
The maximum height \( H \) reached by the projectile is given by:

\[
H = \frac{v_0^2 \sin^2 \theta}{2g}
\]

Let's calculate the range and height.import math

# Given values
v0 = 75  # Initial velocity in m/s
theta = 45  # Angle in degrees
g = 9.8  # Acceleration due to gravity in m/s^2

# Convert angle to radians
theta_rad = math.radians(theta)

# Time of flight formula (not necessary for range and height but useful)
T = (2 * v0 * math.sin(theta_rad)) / g

# Range formula
R = (v0**2 * math.sin(2 * theta_rad)) / g

# Maximum height formula
H = (v0**2 * (math.sin(theta_rad))**2) / (2 * g)

R, H  # Output the range and height in metersThe range of the projectile is approximately \( 574 \, \text{m} \) and the maximum height is approximately \( 143.5 \, \text{m} \).

Would you like more details or have any further questions?

Here are some related questions you might find interesting:
1. How does changing the angle affect the range and height of the projectile?
2. What would happen to the range if the initial velocity was doubled?
3. How does air resistance affect the calculations for range and height?
4. Can we determine the velocity at the maximum height of the projectile?
5. What is the trajectory of the projectile, and how can it be plotted?

**Tip:** The range of a projectile is maximized when it's launched at a 45° angle, assuming no air resistance.

Learn how to calculate the range and maximum height of a projectile fired at an angle of 45° with an initial speed of 0.075km/s. Detailed step-by-step solution using projectile motion equations. Answers provided in meters.

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