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 5.4
m waterfall. The spy’s speed is 14 m/s and
the officials’ speed is 25 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/s^2 .
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 edge and fall.

First, let's find the time it takes for the spy's boat to reach the edge of the waterfall:

We can use the equation of motion:

distance = initial velocity × time + 0.5 × acceleration × time^2

Since the spy's boat has a constant speed of 14 m/s and there is no acceleration in the horizontal direction, the equation simplifies to:

distance = 14 m/s × time

Next, let's find the time it takes for the officials' boat to reach the edge of the waterfall:

Again, using the equation of motion, the distance for the officials' boat can be expressed as:

distance = 25 m/s × time

Since both boats reach the edge of the waterfall at the same time, we can set these two equations equal to each other:

14 m/s × time = 25 m/s × time

We can solve for time:

14 m/s × time = 25 m/s × time
14 m/s - 25 m/s = 0
-11 m/s × time = 0

Since the time cannot be negative, we conclude that the time it takes for both boats to reach the edge of the waterfall is 0 seconds.

Now, let's calculate the distance between the two vessels when they land below the waterfall. We can use the equation of motion for vertical motion, where the initial velocity is 0 m/s (since both boats fall from rest) and the acceleration is the acceleration due to gravity, -9.81 m/s^2:

distance = 0 × time + 0.5 × (-9.81 m/s^2) × time^2

Simplifying the equation:
distance = -4.905 m/s^2 × time^2

Since the time it takes for both boats to reach the edge is 0 seconds, the distance between the two vessels when they land below the waterfall is also 0 meters.

Therefore, when the two boats land below the waterfall, they will be 0 meters apart.

To find the distance between the two vessels when they land below the waterfall, we can calculate the time it takes for the spy and officials' boat to reach the edge of the waterfall, and then calculate the distance each boat travels during that time.

Let's start by calculating the time it takes for each boat to reach the edge of the waterfall.

For the spy's boat:
Initial velocity (u) = 14 m/s
Acceleration (a) = 0 m/s^2 (since the boat's speed is not changing)
Using the formula: s = ut + (1/2)at^2, where s is the distance traveled, t is the time taken:
0 = 14t + (1/2)(0)(t^2)
0 = 14t
t = 0 (This means that the spy's boat reaches the waterfall edge instantaneously.)

For the officials' boat:
Initial velocity (u) = 25 m/s
Acceleration (a) = 0 m/s^2 (since the boat's speed is not changing)
Using the same formula: s = ut + (1/2)at^2
0 = 25t + (1/2)(0)(t^2)
0 = 25t
t = 0 (This also means that the officials' boat reaches the waterfall edge instantaneously.)

As both boats reach the edge of the waterfall at the same time (t = 0), they will land below the waterfall at the same time as well.

Now, let's calculate the distance traveled by each boat during this time.

For the spy's boat:
Distance traveled (s) = ut + (1/2)at^2
s = (14)(0) + (1/2)(0)(0^2)
s = 0

For the officials' boat:
Distance traveled (s) = ut + (1/2)at^2
s = (25)(0) + (1/2)(0)(0^2)
s = 0

Therefore, when they land below the waterfall, the two vessels will be 0 meters apart from each other.