(II) Pelicans tuck their wings and free-fall straight down whe...

Question

(II) Pelicans tuck their wings and free-fall straight down when diving for fish. Suppose a pelican starts its dive from a height of 16.0 m and cannot change its path once committed. If it takes a fish 0.20 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.

Accepted Answer

Hello everyone. Let's go through this practice problem. In a military exercise, bombs are being dropped from a 25 m high bridge. If it takes a regular soldier 0.5 seconds to react and protect themselves from the falling bomb, what's the lowest height at which they need to see the bomb coming to avoid getting hit successfully? A 8.62 m. B 15.1 m C 9.85 m and D 17.2 m. This is a kinematics problem where we have an object. In this case, a bomb that's moving at a presumably constant acceleration. And we're looking to analyze details about it such as the height it travels through and the amount of time it takes as it's moving. Therefore, we're going to want to use our equations for constant acceleration. The main three equations we use in problems like these are the one that says that speed is equal to initial speed plus acceleration multiplied by time displacement is equal to initial speed multiplied by time plus one half acceleration multiplied by time squared. And final speed squared is equal to initial speed squared plus two multiplied by acceleration. Multiplied by displacement. These are the three equations that we encounter most in problems like this. This is a problem that's going to require a couple steps because the problem is asking for the lowest height at which the soldiers need to see the bomb before moving. But we're not even given any information about how long it takes for the ball for the bomb to fall in the first place. So the first thing we're going to do using our equations is calculate the total time required by the bomb to reach the ground. Recall that in these problems, our general approach is to choose the equation that includes the variable we're solving for as well as all of our knowns. So in this case, we're looking for time and we're given information about the displacement being the height of the bridge. So the second equation is going to be the equation that's gonna help us out the most at first because we can use that to find the total time it takes for a bomb to fall 25 m. Using that equation, we can see that the displacement or the height of the bridge of 25 m is equal to the initial speed and multiplied by the time. But the initial speed is going to be zero because if the ball is just dropped from rest, then there is no initial speed. So this term vanishes and plus one half the acceleration which is acceleration due to gravity or 9.81 m per second squared multiplied by the square of the time it takes, which is what we're looking for. If we're solving for time, then we need to do some basic algebra here where we multiply both sides of the equation by two, divide both sides by 9.81 and then take the square root. So T is equal to the square root of two multiplied by 25 m divided by 9.81 m per second squared. If we put that into a calculator, then we find a time value of 2.2576 seconds. Now, according to the way the problem is written, if the soldier wants to avoid being hit by the bomb, they must spot it at 4.5 seconds. So to find the minimum time, we're going to take the total time we just found and subtract 0.5 seconds. And this gives us 1.7576 seconds. So this is how long it takes before the bomb can no longer be spotted. So now we're going to use this time, go back to our kinematics equations and find the vertical distance that the bomb covers. During this time. We're going to use the same equation that we used before. Only. Now we're going to use the variables a little differently this time, it's the height, the displacement that we're trying to find. But now we know the time that we're looking for. So this is equal to the initial speed multiplied by time. But the initial speed again is just zero goes to zero plus one half, multiplied by the gravitational acceleration of 9.81 m per second squared multiplied by the square of the time, which is 1.7576 seconds and don't forget to square it. And if we put that into a calculator, then we find a displacement of 15.1523 m. So that is how far the bomb has fallen by the time it can no longer be spotted. But the problem is phrased in a way that is asking for the lowest height from the perspective of the soldiers on the ground. So we're going to need to actually subtract this because the distance we just found is the distance between the bomb and the bridge. But we want to find the distance between the bomb and the ground. So we're going to take the total height of the bridge 25 m and subtract the height. We just found 15.1523 m, which gives us a height of about 9.85 m. So this is the height from the ground that the bomb can be at before it can no longer be spotted. And that corresponds to option C. So that is our answer to the problem. I hope this video helped you out, please. Consider checking out our other tutoring videos which will give you more experience with these types of problems. That's all for now. And I hope you have a lovely day. Bye bye.

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