Question Video: Relative Speed of Bodies Moving in Opposite Directions Mathematics

A ship was sailing with a uniform velocity directly toward a port that is 144 km away. A patrol aircraft passed over the ship traveling in the opposite direction at 366 km/h. When the aircraft measured the ship’s speed, it appeared to be traveling at 402 km/h. Determine the time required for the ship to reach the port.

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Video Transcript

A ship was sailing with a uniform velocity directly toward a port that is 144 kilometers away. A patrol aircraft passed over the ship, traveling in the opposite direction, at 366 kilometers per hour. When the aircraft measured the ship’s speed, it appeared to be traveling at 402 kilometers per hour. Determine the time required for the ship to reach the port.

Okay, so let’s say that we’ve got this ship moving across the salty sea and headed for port. That port is a distance, we’re told, of 144 kilometers away. And while all this is going on, a patrol aircraft flying in the direction opposite the way the ship is moving passes over the ship and measures the ship’s speed relative to the aircraft to be 402 kilometers per hour. Based on this, we want to solve for the time required for the ship to reach port.

As we start on our solution, let’s recall that when an object moves at a constant speed, we can calculate that speed 𝑠 by dividing the distance the object travels by the time it takes to travel that distance. And note that we can rearrange this equation so that it reads 𝑡 is equal to 𝑑 divided by 𝑠.

In our scenario, it’s a time that we want to solve for. And we’re given a distance. That’s 144 kilometers between the ship and port. But we don’t yet know the speed of our ship. We’ll call that speed 𝑆 sub s. And while we don’t know it, here’s what we do know. Our patrol plane flying in the opposite direction at 366 kilometers per hour perceives the ship to be moving at 402 kilometers per hour. In other words, if we take the speed of the ship 𝑆 sub s and we add it to the real speed of the plane, then we’ll get the perceived speed of the ship relative to the plane.

And note that we add these two values rather than, say, subtract them because the ship and the plane are approaching one another. This equation tells us that if we subtract 366 kilometers per hour from both sides, then we’ll have that 𝑆 sub s equals 402 kilometers per hour minus 366 kilometers an hour, which means the speed of the ship relative to the water is 36 kilometers per hour. And this is the speed that we’ll want to use in our equation to solve for the time required for our ship to reach port.

Now that we know that 𝑆 sub s is 36 kilometers an hour, we can write that the time it takes for our ship to reach port is equal the distance it has to travel divided by 𝑆 sub s. That’s 144 kilometers divided by 36 kilometers per hour. And note that the units of kilometers cancel out, while the units of hours will move up to the numerator. 36 goes into 144 exactly four times. So our answer is that the ship takes four hours to reach port.

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