To determine how far the water balloon has fallen when it is struck by the arrow, we can calculate the distance covered by the balloon during the 5.0-second delay before the arrow is shot. Let's break down the problem:
Calculate the distance fallen by the water balloon during the 5.0-second delay: The distance fallen can be determined using the formula: distance = 0.5 * acceleration * time^2
Here, the acceleration due to gravity can be approximated as -9.8 m/s^2 (negative because it acts downward). Plugging in the values: distance = 0.5 * (-9.8 m/s^2) * (5.0 s)^2
distance = -0.5 * 9.8 m/s^2 * 25 s^2 distance ≈ -122.5 m
The negative sign indicates that the balloon has fallen downward.
Calculate the time it takes for the arrow to reach the height of the balloon: The arrow is shot straight up with an initial velocity of 40 m/s. The time it takes for the arrow to reach its maximum height (the same height as the balloon) can be determined using the formula: time = velocity / acceleration
Here, the acceleration is the acceleration due to gravity (-9.8 m/s^2). Plugging in the values: time = 40 m/s / 9.8 m/s^2 time ≈ 4.08 s
Calculate the distance fallen by the balloon during the time it takes for the arrow to reach the height: Using the same formula as before: distance = 0.5 * acceleration * time^2
Plugging in the values: distance = 0.5 * (-9.8 m/s^2) * (4.08 s)^2 distance ≈ -81.1 m
Again, the negative sign indicates that the balloon has fallen downward.
Calculate the total distance fallen by the balloon when struck by the arrow: Adding the distances calculated in steps 1 and 3: total distance fallen = 122.5 m + 81.1 m total distance fallen ≈ 203.6 m
Therefore, when the arrow strikes the balloon, the water balloon has fallen approximately 203.6 meters.