Video Transcript
A man walked six kilometers in the direction of the east in 1.2 hours. Then he turned and walked eight kilometers in the direction of the north in two hours. Calculate the magnitude of the average velocity of the man.
Let’s begin by reminding ourselves what we mean when we talk about the velocity. Velocity is the rate of change of displacement with respect to time. For that reason, we can calculate the average velocity of a journey by dividing the total displacement by the time taken. Now, of course, displacement is slightly different to distance. It’s direction-dependent. The man walks six kilometers east and then eight kilometers north. The displacement will in fact be the distance between his start point and his endpoint.
So let’s sketch a diagram and see if we can calculate his total displacement. He begins by walking six kilometers east. Then he turns and walks eight kilometers north. This means the angle he makes between the first and second part of his journey must be 90 degrees. This, in turn, means we can construct a right triangle, which will allow us to find his displacement. Let’s label the third side in this triangle 𝑥 kilometers, and this is the length we’re trying to find. Since we have a right triangle for which we know two of its lengths and we’re trying to find the third, we can use the Pythagorean theorem.
This tells us that the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, 𝑥 is the side that sits opposite the right angle. So it’s the hypotenuse. So 𝑥 squared is six squared plus eight squared. The right-hand side becomes 36 plus 64 which is of course equal to 100. So 𝑥 squared is equal to 100. And since 𝑥 is a length, we know it’s going to have a positive value. So we take the square root of both sides. The square root of 100 is 10. So his total displacement is 10 kilometers.
We can calculate his time in hours by finding the sum of the time taken to complete each part of the journey. That’s 1.2 plus two and that’s equal to 3.2 hours. His velocity then in kilometers per hour must be 10 divided by 3.2, giving us a value of 3.125. Correct to one decimal place, that’s equal to 3.1. Now, the question asked us to find the magnitude of the velocity. The reason it’s asked us to find the magnitude is whilst we’ve modeled this as happening in a straight line, displacement has a direction. And we think about the direction as being positive and negative.
The magnitude of the velocity is its size, so whether we have a positive or negative value, the magnitude itself will simply be positive. In this case, it’s already positive, so we leave it as is. But if it was negative, it would indicate an alternative direction. The magnitude of the average velocity of the man then is 3.1 kilometers per hour.
