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Problem 23 Medium Difficulty

SSM As preparation for this problem, review Conceptual Example 10. The drawing shows two planes each about to drop an empty fuel tank. At the moment of release each plane has the same speed of 135 m/s, and each tank is at the same height of 2.00 km above the ground. Although the speeds are the same, the velocities are different at the instant of release, because one plane is flying at an angle of 15.0 above the horizontal and the other is flying at an angle of 15.0 below the horizontal. Find the magnitude and direction of the velocity with which the fuel tank hits the ground if it is from (a) plane A and (b) plane B. In each part, give the directional angles with respect to the horizontal.

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Video Transcript

So the question states that two planes are flying at a 15 degree angle, one above the horizontal and one below the horizontal each at 135 meters per second and a height of 2000 meters, and each plane drops a fuel tank and we're trying to find what the magnitude of the lost he is and the direction of the lost the vector, Um, from the horizontal is of the fuel tank when it hits the ground. So we only really need to solve one of these two problems because this second situation, the fuel tank is going to fly up in the air. And then when it re crosses this horizontal here, the velocity vector is gonna look exactly like E uh, vector we have over here. So I was gonna have a 15 degree angle here, and it's gonna be moving at 135 meters per second. So this movement in the air is not going to affect E the final velocity of the fuel tank when it's the ground. So to solve this first problem, we first need to separate it into components. So we have, um, our velocity into components. We have V of X and V of y so we can figure these out because we know that the co sign of 15 degrees is equal to adjacent overhype Odd news which is Visa Becks over 135 meters per second, which means piece of X is equal to 135 times co side of 15 degrees and the same thing goes for Visa by. But instead of co sign, it will be signed, so we'll have 135 times sign of 15 degrees. So now that we know our initial bossy in the X and Y direction, we can look at the final velocity vector. Soto. Look, something like this would have a horizontal component, a vertical component and then the total magnitude of the velocity. And we know that because this is projectile motion, the horizontal component of the velocity isn't going to change. There's no acceleration. So this will just be 100 35 times co sign of 15 degrees and we need to find Visa y to get the total magnitude of the vector. So to find visa, why we can use our Kinnah Matics equations which state that the final velocity squared is equal to the initial velocity squared, plus two times the acceleration times, the change in displacement. And so we know that the, um, initial velocity in the vertical direction is 135 times sign of 15 degrees. And we also know the acceleration due to gravity is negative 9.8 as well as the displacement is negative 2000 meters. So when we plug in, um, all this will get B squared is equal to, um 135 sign of 15 degrees squared, plus two times negative 9.8 times negative 2000. And just to note, this is actually negative used in the downward direction. But it doesn't really matter because squared. And so when we saw this so we add all this up and then we take the square root, we get that, uh, the bossy in the Y direction. The final velocity is equal to 201 0.5 meters per second. And now that we know this, we confined the A total magnitude which is equal to the square root of piece of X squared plus B supplies squared. And when we plug in our visa. Why? In our visa of ex we get that are total magnitude is equal to 239 0.63 meters per second. And we can also find this angle here which will call for a tow by taking the 10 in verse, which is going to be opposite over adjacent. So the opposite in this case is our visa by and our Jason is Visa vex, and that will be equal to state. And when we plug in our visa by and visa vaccine toe, take the 10 and verse we get, that data is equal to 57 0.3 degrees, and that's the answer.

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Video Transcript

John D. Cutnell, Kenneth W. Johnson

Physics

Elyse Gonzalez

Farnaz Mohseni

Aspen Fenzl

Jared Enns

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Physics

Elyse Gonzalez

Farnaz Mohseni

Aspen Fenzl

Jared Enns

John D. Cutnell, Kenneth W. JohnsonPhysics

Elyse Gonzalez

Farnaz Mohseni

Aspen Fenzl

Jared Enns

John D. Cutnell, Kenneth W. Johnson

Physics