A surface ship is moving (horizontally) in a straight line at ...

Question

A surface ship is moving (horizontally) in a straight line at 10 km/hr. At the same time, an enemy submarine maintains a position directly below the ship while diving at an angle that is 20° below the horizontal. How fast is the submarine’s altitude decreasing?

Accepted Answer

Welcome back, everyone. In this problem, a drone is flying horizontally at a constant speed of 15 m per second. simultaneously, sorry, a bird is flying directly above the drone descending at an angle of 30 degrees below the horizontal. How quickly is the bird losing altitude? A says it's 15 routes 3 m per second, B 15 m per second, C 5 route 3 m per second, and D 5 m per second. Now, let's try to draw what's going on here so we can better understand and that can help us to solve. OK? So we're saying that we have a bird, OK. We have a bird, straw bird like this, and then we have a drone, you know, it has the four sides, OK. And we know that our drone is flying at a constant horizontal speed of 15 m per second. Now for a bird, we also know that it's flying, it's descending at an angle of 30 degrees below the horizontal. So if we have our horizontal here, then we have the bird's movement and we know that it will also be losing altitude, OK? So this is gonna be 30 degrees, OK, and we can call this speed here VB. OK. And now, we know the way it's losing altitude will be the vertical component of VB's call it VBY, and it also has a horizontal component, VBX. Now how can we use that to figure out how quickly the bird is losing altitude? In other words, how can we use it to find VBY? Well, let's first think about what we know. So far we know that the drone, the drone speed, let's call it VD, is 15 m per second. We know that the angle of descent of the bird theta is 30 degrees, and we already know that the vertical speed is VBY. Since the bird is descending at a 30 degree angle, the vertical velocity has a negative direction in a coordinate system, OK? And the magnitude of the vertical component is what we need that represents the rate of altitude loss. Now, if we think about it here, since the vertical component VBY, OK. Is opposite. So VB, OK. Then because this forms, because the horizontal and vertical components form a right angle, a right angle triangle is made to represent the bird's descent, where the vertical component VB Y will be equal to VB multiplied by sine theta. On the other hand, since the bird is flying directly above the drone and descending at an angle of 30 degrees, its horizontal speed must match the drone's speed. In other words, VBX is going to be. Eal to VD, OK, which is VB class the. Now, we know, OK, if we substitute our values here that VD is 15 m per second, that's the speed of the drone. So if I put this in red, this implies that VB because theta is equal to 50 m per second. I theta is 30 degrees. Which means that VB is gonna be equal to 15 m per second divided by the cosine of 30 degrees. If that's the case, we can use that to help us figure out the. Vertical component of VB, OK, because by substitution, then that means that VBY is gonna be equal to VB 15 m per second divided by the cosine of 30 degrees, OK, multiplied by the sign of theta, the sign of 30 degrees. Now we know that the cosine of 30 degrees is a square root of 3 divided by 2, and we know that the sign of 30 degrees is a 1/2. If we simplify here. Then we could rewrite this as 15 m per second divided by route 3. And now we're going to be multiplying by route 3 divided by route 3 to rationalize, OK? And thus when we do that, we get our vertical component of VB to be equal to 5 route 3 m per second. Thus, the bird is losing altitude at a rate of 5 route 3 m per second, which means C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

A surface ship is moving (horizontally) in a straight line at 10 km/hr. At the same time, an enemy submarine maintains a position directly below the ship while diving at an angle that is 20° below the horizontal. How fast is the submarine’s altitude decreasing?

Key Concepts

Related Rates

Trigonometric Functions

Velocity Components

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