Video Transcript
A cyclist joins an east–west road and travels a distance of 240 meters east at a speed of 12 meters per second. He then immediately turns around and travels a distance of 96 meters west at the same speed that he traveled east at and then stops to dismount his bike. What is the total time for which the cyclist moves? And what is the cyclist’s average velocity to the east from when he joins the road to when he dismounts? Give your answer to two decimal places.
Before we tackle the question itself, let’s quickly draw out how the cyclist’s journey looks like. He joins the road at some point and first travels east along it for a distance of 240 meters. He does this at a speed of 12 meters per second. He then travels back in the opposite direction west along the road for a distance of 96 meters. Again, he does this at the same speed that is 12 meters per second. At this point, he stops and dismounts the bike.
The first part of the question is asking us for the total time for which the cyclist moves. Since we’re told that the cyclist immediately turns around between traveling east and traveling west, we can assume that he takes no time to change direction. In other words, he changes direction instantaneously. Then the total time for which he is moving is given by the sum of the time for which he is moving to the east, which we’ll call 𝑡 subscript e and the time for which he is moving to the west, 𝑡 subscript w.
To calculate these times, we need to recall the definition of speed, 𝑠, as distance traveled, 𝑑, divided by time taken, 𝑡. We can rearrange this to give that the time 𝑡 is given by the distance 𝑑 divided by the speed 𝑠. To calculate 𝑡 subscript e, we note that the question has told us the distance traveled to the east is 240 meters and the speed is 12 meters per second. Thus, we can write 𝑑 subscript e equals 240 meters and 𝑠 subscript e equals 12 meters per second. We then need to substitute those numbers into our equation for the time.
We get that 𝑡 subscript e is equal to 240 divided by 12. This is equal to 20 seconds. Note that we get units of seconds because the distance was in meters and the speed was in meters per second. Now for 𝑡 subscript w, we’re told that the speed is the same as a speed at which he travels east, i.e., 12 meters per second. Recall that speed is a scalar quantity; it has only a magnitude, and there is no direction associated with it. So we don’t need to worry that our cyclist is now travelling the opposite direction. His speed, 𝑠 subscript w, is still the same at 12 meters per second.
This time, we’re given that the distance is 96 meters. We therefore have 𝑑 subscript w equals 96 meters. Again, distance is also a scalar quantity, so we don’t need to worry about the direction. Substituting in these numbers into our time equation gives us that 𝑡 subscript w is given by 96 meters divided by 12 meters per second. This gives 𝑡 subscript w is equal to eight seconds. Recall that the question asked us for the total time for which the cyclist moves and that we said that this is the sum of the time for which he travels to the east and the time for which he travels to the west. So our answer to the first part of the question is given by the sum of 𝑡 subscript e plus 𝑡 subscript w, that is, 20 seconds plus eight seconds. This gives us a total time of 28 seconds.
The second part of the question is asking for the cyclist’s average velocity to the east from when he joins the road to when he dismounts. Recall that average velocity is defined as the total displacement divided by the total time taken. The total displacement is the shortest distance between the start point and the endpoint. Note that velocity is a vector quantity; it has a direction as well as a magnitude.
Since the cyclist only travels east and west, we know that the average velocity must lie along this east–west line, i.e, the direction of it must either be east or west. The question asks us for the average velocity to the east. We can see from our diagram that the cyclist travels further to the east than he does back to the west. This means that his overall displacement will be to the east. So we’re expecting a positive number of his average velocity to the east. Note that if he moved to the west overall, his average velocity would be in the westward direction, which we could write as a negative velocity to the east.
On our diagram, this total displacement is given by this distance here, the shortest distance between the two x’s representing the start point and the endpoint of his motion. He travels 240 meters east and then 96 meters west. Therefore, his total displacement to the east is 240 minus 96. This is 144 meters. The total time he takes to do this is the sum of the time for which he is traveling to the east, 𝑡 subscript e, and the time for which he’s traveling to the west, 𝑡 subscript w.
We already calculated this sum in the first part of the question. We therefore know that the total time is 28 seconds. Therefore, the average velocity to the east is given by the displacement, 144 meters, divided by the time, 28 seconds. We’re asked to give the answer to two decimal places. To this precision, we have our answer as 5.14 meters per second. Note that we do not need to specify direction as the question has specified that we’re given the average velocity to the east.
