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To solve this projectile motion problem, we need to break down the motion into horizontal and vertical components. Let's begin with the provided dataTo solve this projectile motion problem, we need to break down the motion into horizontal and vertical components. Let's begin with the provided 

- Initial velocity \( v_0 = 49.5 \, \text{m/s} \)
- Launch angle \( \theta = 34.5^\circ \)
- Time of flight \( t = 2.34 \, \text{s} \)
  
### Horizontal Distance (\( d_H \)):
The horizontal distance is determined by the horizontal component of the velocity. The horizontal velocity \( v_{0x} \) can be found using:

\[
v_{0x} = v_0 \cdot \cos(\theta)
\]

The horizontal distance is given by:

\[
d_H = v_{0x} \cdot t
\]

### Vertical Distance (\( d_V \)):
The vertical distance is determined by the vertical component of the velocity. The vertical velocity \( v_{0y} \) is:

\[
v_{0y} = v_0 \cdot \sin(\theta)
\]

The vertical distance traveled by the projectile at time \( t \) can be calculated using the equation for vertical displacement under constant acceleration (due to gravity):

\[
d_V = v_{0y} \cdot t - \frac{1}{2} g t^2
\]

where \( g = 9.8 \, \text{m/s}^2 \) is the acceleration due to gravity.

Let's calculate these values.The results are as follows:

- The horizontal distance \( d_H \) is approximately **95.46 meters**.
- The vertical distance \( d_V \) is approximately **38.78 meters**.

Do you have any questions or would you like further details?

Here are 5 related questions to expand on this topic:
1. How would the time of flight change if the launch angle was increased?
2. What would the horizontal and vertical distances be if the projectile were launched at a different speed?
3. How does air resistance affect the range and height of a projectile in real-world scenarios?
4. How can we determine the maximum height reached by the projectile in this problem?
5. What is the effect of gravity on the horizontal motion of the projectile?

**Tip:** In projectile motion, the horizontal and vertical motions are independent of each other except for the common time of flight.

A projectile is launched from ground level with an initial speed of 49.5 m/s at an angle of 34.5° above the horizontal. It strikes a target in the air 2.34 s later. What is the horizontal distance dH from where the projectile was launched to where it hits the target? What is the vertical distance dV from where the projectile was launched to where it hits the target?

Solve a projectile motion problem involving a projectile launched with an initial speed of 49.5 m/s at an angle of 34.5° above the horizontal. Calculate both the horizontal and vertical distances traveled before hitting the target after 2.34 seconds.

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