Calculate the average velocity of a car which moved in a straight line a distance of 120 meters with a velocity of eight meters per second and then moved in the same direction on the same line a distance of 180 meters with a velocity of six meters per second. Give your answer in meters per second to one decimal place.
In order to answer this question, we need to remind ourselves of the formula we use to calculate average velocity. The average velocity of an object is found by dividing the total displacement by the total time, where of course displacement is how far the object has traveled from its starting point or some frame of reference. Now, in this case, there are actually two parts to the journey. Over the first part of the journey, the car travels 120 meters at a velocity of eight meters per second. Then it travels a farther 180 meters, but this time with a velocity of six meters per second. Now it’s traveling in the same direction on a straight line, so we can work out the total displacement and the time that the car takes to travel this far.
Let’s define 𝑑 sub one as being the distance the car travels in the first part of the journey. That’s 120 meters. And 𝑣 sub one is its velocity, eight meters per second. Substituting what we know about this part of the journey into the formula and we get eight equals 120 over 𝑡 sub one. Let’s multiply by 𝑡 sub one, and we get eight 𝑡 sub one equals 120. Then we divide through by eight, and we get 𝑡 sub one equals 120 divided by eight. That’s 15. So, we found that over the first part of the journey, the car travels 120 meters and it takes 15 seconds.
Let’s repeat this for the second part of the journey. This time, we’ll define 𝑑 sub two to be 180 meters and 𝑣 sub two, the velocity, to be six meters per second. Substituting into the formula once again and we get six equals 180 divided by 𝑡 sub two. Rearranging as before, in other words, multiplying by 𝑡 sub two and then dividing by six, and we get 𝑡 sub two is 180 divided by six, or 30. So, over the second part of the journey, the car traveled 180 meters and it took 30 seconds. We’re now ready to calculate the total displacement of the car and the total time.
Now remember, the car is traveling in a single straight line in the same direction. So, the total displacement will be 120 meters plus 180 meters, and that’s 300 meters. Similarly, the total time will be 15 seconds add 30 seconds, and that’s 45 seconds. So, the average velocity over the entire journey must therefore be 300 divided by 45. And that’s 6.6 recurring. Now correct to one decimal place and in meters per second then, we have shown that the average velocity of the car is 6.7 meters per second.
Now, contextually, this makes a lot of sense. The car traveled at some distance for eight meters per second and then at another distance for six meters per second. It would make sense that the average velocity then is somewhere between these two values, so 6.7 meters per second seemed very reasonable.