Find the maximum height H if the projectile is fired at an angle of 

45 degrees45° 

to the horizontal with an initial speed of 

160160 

feet per second.




To find the maximum height \( H \) of the projectile when it is fired at an angle of \( \theta = 45^\circ \) with an initial speed of \( v_0 = 160 \) feet per second, we use the formula for maximum height:

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

### Step 1: Plug in the given values
- \( v_0 = 160 \) feet per second
- \( \theta = 45^\circ \)
- \( g = 32.2 \) feet per second squared

### Step 2: Simplify the trigonometric function
Since \( \theta = 45^\circ \), we know:
\[
\sin 45^\circ = \frac{\sqrt{2}}{2}
\]

So,
\[
\sin^2 45^\circ = \left(\frac{\sqrt{2}}{2}\right)^2 = \frac{2}{4} = \frac{1}{2}
\]

### Step 3: Substitute into the formula
\[
H = \frac{(160)^2 \times \frac{1}{2}}{2 \times 32.2}
\]

### Step 4: Simplify and compute
First, calculate \( 160^2 \):
\[
160^2 = 25600
\]

Now, calculate the height:
\[
H = \frac{25600 \times \frac{1}{2}}{64.4} = \frac{12800}{64.4} \approx 198.76 \text{ feet}
\]

So, the maximum height \( H \) is approximately **198.76 feet**.

Would you like any further explanation or have other questions?

Here are some related questions to consider:

1. How would the maximum height change if the initial speed was increased to 200 feet per second?
2. What is the time taken to reach this maximum height?
3. How does changing the angle of projection to \(30^\circ\) affect the maximum height?
4. What would be the maximum height if the gravity \( g \) was 9.8 m/s\(^2\) (standard on Earth)?
5. How does air resistance affect the maximum height in real-life scenarios?

**Tip:** The maximum height is always reached when the vertical component of the velocity becomes zero.

Learn how to find the maximum height of a projectile fired at an angle of 45 degrees with an initial speed of 160 feet per second using projectile motion formulas and trigonometry. Get a step-by-step solution and understand the calculations involved.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation