Pelicans tuck their wings and free-fall straight down when diving for fish. Suppose a pelican starts its dive from a height of 15.0 m and cannot change its path once committed.
If it takes a fish 0.25 s to perform evasive action, at what minimum height must it spot the pelican to escape? Assume the fish is at the surface of the water.
The fish must spot the pelican when it is 0.25 s (or more) from hitting the water.
The pelican takes
t = sqrt(2H/g) = 1.75 s to hit the water if it starts its dive at H = 15 m.
Its vertical drop 1.50 s after starting the dive is y = (1/2)g*(1.50)^2 = 11.03 m
It is 15 - 11.03 = 4.97 m above the water surface at that time,
Well, this is quite a fishy situation, isn't it? The pelican is diving for its lunch, and the poor fish has to come up with an escape plan. Talk about pressure!
Now, let's do some calculations and figure out how high the fish needs to spot the pelican to escape. We know that the pelican takes 0.25 seconds to perform evasive action. During this time, the pelican is in free fall and will travel a certain distance before reaching the water.
To find the distance traveled by the pelican, we need to apply the laws of physics. Using the equation d = 0.5gt^2, where d is the distance, g is the acceleration due to gravity (approximately 9.8 m/s^2), and t is the time, we can solve for d.
Since the pelican starts its dive from a height of 15.0 m, the distance traveled by the pelican will be (15.0 - d). This is the distance the fish needs to spot the pelican to have a fighting chance.
So, let's plug in the numbers and calculate:
15.0 - d = 0.5 * 9.8 * (0.25)^2
Now, if my calculations are as sharp as a pelican's beak, the minimum height that the fish needs to spot the pelican to escape is the value of d that solves this equation. I'll leave you with the honors of crunching those numbers and finding the fish's ticket to freedom. Good luck!
To determine the minimum height at which the fish must spot the pelican to escape, we can use the equations of motion. We'll assume that the initial velocity of the pelican is zero when it starts its dive.
The equation for displacement (d) under constant acceleration (a) and time (t) can be given as:
d = ut + (1/2)at^2
Since the initial velocity of the pelican is zero, the equation reduces to:
d = (1/2)at^2
We can rearrange the equation to solve for acceleration (a):
a = (2d) / t^2
In this case, the distance the pelican needs to travel downward is 15.0 m, and the time it takes for the fish to perform evasive action is 0.25 s.
Substituting the given values:
a = (2 * 15.0 m) / (0.25 s)^2
Calculating:
a = (30.0 m) / (0.0625 s^2)
a = 480 m/s^2
Now, let's consider the fish's acceleration to be zero since it is performing no evasive action at first. The distance covered by the fish when it starts to react (d_fish) can be calculated using the equation:
d_fish = (1/2) * a_fish * t^2
where a_fish is the acceleration of the fish and t is the time it takes for the fish to perform evasive action.
Since the fish's acceleration and the pelican's acceleration are the same (since they both experience the same gravitational force), we can substitute the calculated acceleration (a) into the equation:
d_fish = (1/2) * 480 m/s^2 * (0.25 s)^2
Calculating:
d_fish = (1/2) * 480 m/s^2 * 0.0625 s^2
d_fish = 15 m
Therefore, the fish must spot the pelican at a minimum height of 15.0 meters to escape by performing evasive action.
To determine the minimum height from which the fish must spot the pelican to escape, we need to consider the time it takes for the pelican to reach the water surface.
Let's analyze the situation step by step:
1. Find the time it takes for the pelican to reach the water surface:
Since the pelican is free-falling vertically, we can use the equation for the time it takes an object to fall a certain distance in free fall:
h = (1/2) * g * t^2
where:
h = height (15.0 m)
g = acceleration due to gravity (approximately 9.8 m/s^2)
t = time
Rearranging the equation to solve for t:
t^2 = 2h/g
Substituting the given values:
t^2 = (2 * 15.0 m) / (9.8 m/s^2)
Simplifying:
t^2 = 30.0 m / 9.8 m/s^2
t^2 ≈ 3.06 s^2
Taking the square root of both sides:
t ≈ √3.06 s
t ≈ 1.75 s
So, it takes approximately 1.75 seconds for the pelican to reach the water surface.
2. Find the distance the fish can cover in 0.25 seconds:
Given that the fish has 0.25 seconds to perform evasive action, we want to calculate how far it can swim within that time frame.
Using the equation for distance traveled with uniform motion:
d = v * t
where:
d = distance
v = velocity
t = time (0.25 s)
The velocity of the fish can be assumed to be constant.
3. Determine the corresponding height:
Since the fish needs to spot the pelican and swim away before the pelican reaches the water, the fish's distance traveled should be greater than or equal to the height from which the pelican starts its dive.
So, the height from which the fish needs to spot the pelican can be calculated by rearranging the equation:
h = d
Therefore, the minimum height from which the fish must spot the pelican to escape is equal to the distance traveled by the fish in 0.25 seconds.
Calculating the distance:
d = v * t
Given that t = 0.25 s, the time the fish has to escape:
d = v * 0.25 s
To determine the minimum height, we need to find the maximum velocity the fish can achieve in that time period.
Since the problem does not provide the velocity of the fish, we cannot calculate the minimum height required without additional information.
Hence, without knowing the fish's maximum velocity, we cannot determine the minimum height it would need to spot the pelican to escape.