To find out how far the dart would land when shot horizontally from a height of 3.0 meters, we can use the concepts of projectile motion.
1. Time of Flight (t): When an object is shot horizontally, it will fall to the ground due to gravity. The time it takes to fall can be calculated using the formula for the distance fallen under uniform acceleration due to gravity:
\[ h = \frac{1}{2} g t^2 \]
Where:
- \( h \) is the height (3.0 m),
- \( g \) is the acceleration due to gravity (approximately \( 9.81 , \text{m/s}^2 \)).
Rearranging the formula to solve for \( t \):
\[ t^2 = \frac{2h}{g} \] \[ t = \sqrt{\frac{2h}{g}} = \sqrt{\frac{2 \times 3.0 , \text{m}}{9.81 , \text{m/s}^2}} \]
Calculating it:
\[ t = \sqrt{\frac{6.0}{9.81}} \approx \sqrt{0.612} \approx 0.782 , \text{s} \]
2. Horizontal Distance (d): The horizontal distance traveled can be found using the formula:
\[ d = v \cdot t \]
Where:
- \( v \) is the horizontal velocity (14.5 m/s),
- \( t \) is the time of flight we just calculated.
Now substituting the values:
\[ d = 14.5 , \text{m/s} \times 0.782 , \text{s} \approx 11.34 , \text{m} \]
Therefore, the dart would land approximately 11.34 meters from the point where it was shot horizontally.