To calculate the horizontal distance traveled by a cannonball fired at an initial velocity of 85 m/s at an angle of 45° above the horizontal, we can use the following steps:
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Resolve the initial velocity into horizontal and vertical components:
- The horizontal component (\(v_{x0}\)) is given by: \[ v_{x0} = v_0 \cdot \cos(\theta) \]
- The vertical component (\(v_{y0}\)) is given by: \[ v_{y0} = v_0 \cdot \sin(\theta) \]
where \(v_0 = 85 , \text{m/s}\) and \(\theta = 45°\).
Calculating these components:
- Since \(\cos(45°) = \sin(45°) = \frac{\sqrt{2}}{2}\): \[ v_{x0} = 85 \cdot \frac{\sqrt{2}}{2} \approx 60.1 , \text{m/s} \] \[ v_{y0} = 85 \cdot \frac{\sqrt{2}}{2} \approx 60.1 , \text{m/s} \]
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Determine the time of flight: The time of flight (\(t\)) for a projectile can be found using the formula: \[ t = \frac{2 \cdot v_{y0}}{g} \] where \(g \approx 9.81 , \text{m/s}^2\) is the acceleration due to gravity.
Substituting in the values: \[ t = \frac{2 \cdot 60.1}{9.81} \approx \frac{120.2}{9.81} \approx 12.24 , \text{s} \]
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Calculate the horizontal distance traveled: The horizontal distance (\(d\)) can be calculated using: \[ d = v_{x0} \cdot t \] Substituting in the values: \[ d = 60.1 \cdot 12.24 \approx 735.7 , \text{m} \]
Therefore, the horizontal distance traveled during the flight is approximately 735.7 meters.
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