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
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.