To find the initial height of a balloon from which a bomb is dropped, we first need to understand the situation and the relevant formulas. Here's a step-by-step guide to calculating the initial height:
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Variables:
- Let \( h_0 \) be the initial height of the balloon above the ground.
- Let \( v_b \) be the balloon's ascent speed (in meters per second).
- Let \( t \) be the time it takes for the bomb to hit the ground (in seconds).
- Let \( g \) be the acceleration due to gravity, approximately \( 9.81 , \text{m/s}^2 \).
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Height of the Balloon When Bomb is Dropped:
- During the time \( t \) that the bomb is falling, the balloon continues to rise. The height of the balloon when the bomb is dropped can be calculated as: \[ h_{\text{balloon}} = h_0 + v_b \cdot t \]
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Free Fall of the Bomb:
- When the bomb is dropped, it falls from the height \( h_{\text{balloon}} \) under the influence of gravity. The formula for the distance \( d \) the bomb falls after \( t \) seconds is: \[ d = \frac{1}{2} g t^2 \]
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Setting Up the Equation:
- The bomb hits the ground, which can be set as the height equal to zero \( (h = 0) \). Thus, at the time \( t \): \[ h_{\text{balloon}} - d = 0 \]
- Substituting for \( h_{\text{balloon}} \) and \( d \), we have: \[ h_0 + v_b \cdot t - \frac{1}{2} g t^2 = 0 \]
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Solving for Initial Height \( h_0 \):
- Rearranging the equation gives: \[ h_0 = \frac{1}{2} g t^2 - v_b \cdot t \]
Now you can plug in the values for \( g \), \( v_b \), and \( t \) to calculate the initial height \( h_0 \) of the balloon from which the bomb was dropped.
Example Calculation: If the balloon's ascent speed \( v_b = 5 , \text{m/s} \) and the bomb takes \( t = 2 , \text{s} \) to hit the ground, you can calculate the height as follows:
- Compute \( d = \frac{1}{2} (9.81) (2^2) = \frac{1}{2} \times 9.81 \times 4 = 19.62 , \text{m} \).
- Compute \( h_0 = 19.62 , \text{m} - (5 \cdot 2) = 19.62 - 10 = 9.62 , \text{m} \).
Thus, the initial height of the balloon would be approximately \( 9.62 , \text{m} \).
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