A spy in a speed boat is being chased down a
river by government officials in a faster craft.
Just as the officials’ boat pulls up next to the
spy’s boat, both boats reach the edge of a 4.6
m waterfall. The spy’s speed is 15 m/s and
the officials’ speed is 24 m/s.
How far apart will the two vessels be when
they land below the waterfall? The accelera-
tion of gravity is 9.81 m/s2 .
Answer in units of m
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To find the distance between the two vessels when they land below the waterfall, we need to calculate the time it takes for each boat to reach the bottom of the waterfall.
Let's first find the time it takes for the spy's boat to reach the bottom of the waterfall. We can use the equation of motion:
distance = (initial velocity * time) + (0.5 * acceleration * time^2)
Since the acceleration is due to gravity, we can substitute accleration = 9.81 m/s^2, initial velocity = 15 m/s and distance = 4.6 m. We can rearrange the equation to solve for time:
4.6 m = (15 m/s) * t + 0.5 * (9.81 m/s^2) * t^2
Simplifying the equation:
0.5 * (9.81 m/s^2) * t^2 + (15 m/s) * t - 4.6 m = 0
Solving this quadratic equation will give us the time it takes for the spy's boat to reach the bottom of the waterfall.
Using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
a = 0.5 * (9.81 m/s^2) = 4.905 m/s^2
b = (15 m/s)
c = -4.6 m
Plugging in these values:
t = (-15 m/s ± √((15 m/s)^2 - 4 * 4.905 m/s^2 * (-4.6 m))) / (2 * 4.905 m/s^2)
Solving this equation will give us two values for time, but we'll only consider the positive value since time can't be negative.
Calculate the time taken by the spy's boat:
t = (-15 m/s + √((15 m/s)^2 - 4 * 4.905 m/s^2 * (-4.6 m))) / (2 * 4.905 m/s^2)
t = 0.918 s
Now that we have the time it takes for the spy's boat, we can find the distance it covers during this time:
distance = (initial velocity * time)
distance = (15 m/s) * (0.918 s)
Calculate the distance covered by the spy's boat:
distance = 13.77 m
Since the officials' boat reaches the edge of the waterfall at the same time as the spy's boat, they will be 13.77 meters apart when they land below the waterfall.