A projectile is fired from an initial height of 101m above the ground with an initial velocity of 111m/s and at an angle of 38∘ above the horizontal. Determine the projectile's speed when it reaches the ground. Use exactly 9.8m/s^2 for the acceleration due to gravity, and round your answer to the nearest meter.

To solve this problem, we need to determine the speed of the projectile when it reaches the ground. This requires us to break the motion into two components: horizontal and vertical.

### Step 1: Decompose the Initial Velocity
The initial velocity of 111 m/s is directed at an angle of 38° above the horizontal. We can split it into horizontal (\( v_{0x} \)) and vertical (\( v_{0y} \)) components using trigonometric functions:

- \( v_{0x} = v_0 \cos(\theta) = 111 \cos(38^\circ) \)
- \( v_{0y} = v_0 \sin(\theta) = 111 \sin(38^\circ) \)

### Step 2: Time of Flight
The time of flight can be found by solving the vertical motion. We use the kinematic equation for vertical displacement (\( y = v_{0y} t - \frac{1}{2} g t^2 \)):

\[
y_f = y_0 + v_{0y} t - \frac{1}{2} g t^2
\]
Where:
- \( y_f = 0 \) (the final height when it hits the ground)
- \( y_0 = 101 \) m (the initial height)
- \( v_{0y} \) is the initial vertical velocity
- \( g = 9.8 \, \text{m/s}^2 \)

### Step 3: Vertical Final Velocity
The vertical velocity at any point is given by the equation:
\[
v_y = v_{0y} - g t
\]

At the point of impact, the final vertical velocity will contribute to the total speed.

### Step 4: Total Speed at Ground
The final speed of the projectile when it reaches the ground is the combination of both horizontal and vertical components at that instant:
\[
v = \sqrt{v_x^2 + v_y^2}
\]
Where:
- \( v_x \) is the constant horizontal velocity
- \( v_y \) is the vertical velocity at the ground

I'll calculate this now.The projectile's speed when it reaches the ground is approximately **120 m/s**.

Would you like further details or explanations? Here are some related questions you might find interesting:

1. How does the launch angle affect the horizontal and vertical components of velocity?
2. What would happen to the final speed if the initial height were different?
3. How do we determine the total flight time in projectile motion problems?
4. Can we solve this problem using energy conservation principles instead of kinematics?
5. How would air resistance affect the projectile's motion and final speed?

**Tip:** When solving projectile motion problems, always separate the horizontal and vertical components of motion, as they are influenced differently by gravity.

Solve a projectile motion problem where a projectile is fired from a height of 101m with a velocity of 111m/s at an angle of 38°. Learn how to decompose velocity, calculate flight time, and determine the final speed using kinematic equations. Perfect for high school students studying Grades 10-12 physics.

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