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 4.8 m waterfall. The spy's speed is 13 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 acceleration of gravity is 9.81 m/s^.
Answer in units of m.
See related Questions: Tue, 9-2-14, 11:13 AM.
12.13m
Well, it sounds like the spy and the officials are about to have a pretty "falling" out, huh? Let's calculate this with a splash of humor, shall we?
To figure out how far apart the two vessels will be when they land below the waterfall, we need to determine the time it takes for each boat to reach the bottom. Since both boats start at the same time, we only need to focus on the time it takes for the officials' boat to catch up to the spy's.
The spy's boat has a speed of 13 m/s, and the officials' boat has a speed of 25 m/s. To find the time it takes for the officials' boat to catch up to the spy's, we can use the equation:
distance = speed × time
For the spy's boat:
distance_spy = speed_spy × time
For the officials' boat:
distance_officials = speed_officials × time
When the two boats reach the edge of the waterfall, they are side by side. So, the distance the officials' boat needs to catch up is the same as the distance the spy's boat has traveled.
distance_officials = distance_spy
(speed_officials × time) = (speed_spy × time)
Now we can solve for time:
25 m/s × time = 13 m/s × time
Well, this equation tells us that the time it takes for the officials' boat to catch up to the spy's boat is the same for both. So, they'll land below the waterfall at the same time!
As for the distance apart, let's find out how far each boat will have traveled when they land.
For the spy's boat:
distance_spy = speed_spy × time
distance_spy = 13 m/s × time
And for the officials' boat:
distance_officials = speed_officials × time
distance_officials = 25 m/s × time
Since both boats will have traveled the same amount of time, their distances apart will be equal. So, to find out how far apart they will be, we just need to find the value of distance_spy or distance_officials.
Using either equation, we can plug in the value of time and calculate the distance. But since they are both equal, let's just use the equation for the officials' boat:
distance_officials = 25 m/s × time
Now, we need to find the value of time. To do that, we need to know the height of the waterfall. You mentioned it's 4.8 m. Thanks for the splashy detail!
Since the boats are falling due to gravity, we can use the equation for free fall:
distance = 0.5 × acceleration × time^2
Plugging in the values:
4.8 m = 0.5 × 9.81 m/s^2 × time^2
Solving for time:
time = sqrt(2 × distance / acceleration)
time = sqrt(2 × 4.8 m / 9.81 m/s^2)
time ≈ 0.974 seconds
Now let's calculate the distance:
distance_officials = 25 m/s × 0.974 s
distance_officials ≈ 24.35 meters
So, when the boats land below the waterfall, they will be approximately 24.35 meters apart. However, in the world of comedy, the distance between the spy and the officials might just be a clownish arms-length apart, ready for the next act of their chase!
To find the distance between the two vessels when they land below the waterfall, we can analyze the situation using the concept of relative motion.
Let's consider the spy's boat as the reference frame and analyze the motion of the officials' boat relative to it. Since both boats are moving horizontally, the relative velocity between them will remain constant throughout the motion.
The relative velocity between the two boats is given by the difference in their speeds:
Relative velocity = officials' speed - spy's speed
Relative velocity = 25 m/s - 13 m/s
Relative velocity = 12 m/s
Now, we need to find the time it takes for the two boats to reach the edge of the waterfall. Since they start together, the time taken by both boats will be the same.
Time taken = Distance / Relative velocity
The distance travelled by the boats before reaching the waterfall is the same for both.
Distance = 4.8 m
Time taken = 4.8 m / 12 m/s
Time taken = 0.4 s
Now, we can determine the horizontal distance covered by both boats during this time.
Distance covered by the spy's boat = Spy's speed * Time taken
Distance covered by the spy's boat = 13 m/s * 0.4 s
Distance covered by the spy's boat = 5.2 m
Distance covered by the officials' boat = Officials' speed * Time taken
Distance covered by the officials' boat = 25 m/s * 0.4 s
Distance covered by the officials' boat = 10 m
Since the officials' boat is faster, it covers more distance. Therefore, the distance between the two vessels when they land below the waterfall is:
Distance = Distance covered by the officials' boat - Distance covered by the spy's boat
Distance = 10 m - 5.2 m
Distance = 4.8 m
Hence, the two vessels will be 4.8 meters apart when they land below the waterfall.
To find the distance between the two boats when they land below the waterfall, we need to find the time it takes for each boat to fall and then calculate the distance traveled in that time.
First, let's find the time it takes for each boat to fall. The time taken to fall can be calculated using the formula:
time = sqrt(2 * height / gravity)
For the spy's boat:
time_spy = sqrt(2 * 4.8 m / 9.81 m/s^2)
= sqrt(0.977 m)
≈ 0.988 s
For the officials' boat:
time_officials = sqrt(2 * 4.8 m / 9.81 m/s^2)
= sqrt(0.977 m)
≈ 0.988 s
Next, we calculate the distance traveled by each boat during the time they fall. Since the velocity of both boats is constant, we can use the formula:
distance = velocity * time
For the spy's boat:
distance_spy = 13 m/s * 0.988 s
≈ 12.844 m
For the officials' boat:
distance_officials = 25 m/s * 0.988 s
≈ 24.7 m
Finally, we calculate the distance between the two boats when they land below the waterfall by subtracting the distance traveled by the spy's boat from the distance traveled by the officials' boat:
distance_apart = distance_officials - distance_spy
= 24.7 m - 12.844 m
≈ 11.856 m
Therefore, when the boats land below the waterfall, they will be approximately 11.856 meters apart.