(a) To find the distance traveled, you would add up the distances that were traveled in each leg of the journey. Since Distance = Rate x Time ( d= rt), we can calculate the distance traveled in the first leg, going south, as 1.5 x 6.0, giving us 9.0 km. We are told the distance traveled on the second leg: 4.0 km. So, the total distance equals 9.0 plus 4.0, or 13 km.
(b) To find the displacement, we remember that displacement equals the distance they ended up from the starting point...drawing a straight line from their starting point to where they left off. Since they went 9.0 km south (down) and then 4.0 km east (right), those would be the lengths of two sides of a right triangle. Then, connecting their starting point to their ending point, you would be drawing the hypotenuse of a right triangle. We can calculate the length of the hypotenuse, when we know the lengths of the two legs, by using the Pythagorean theorem. Using that, we would get the formula...
9.0^2 + 4.0^2 = d^2, where d is the length of the hypotenuse, which equals the distance from the starting point, or the displacement. Solving for d, we would get d = 9.9 km, rounding to two decimal places.
(c) To calculate average speed, we remember that it equals the total distance traveled divided by the total time. Well, we can't calculate the total time yet, as we don't know the time traveled on the second leg of the journey. Well, we once again use the distance formula (d = rt) to calculate it. Substituting in the distance and rate, we get...
4.0 = 8.0t
Solving for t, we determine that t equals 0.50 hours. So, the total time of the trip equals 1.5 hours + 0.50 hours, which comes out to 2.0 hours.
(d) To calculate average velocity, we remember that it will equal the displacement divided by the total time, and it will also include a direction component. Well, displacement / total time would be 9.9 km / 2.0 hours, which comes out to 4.95 km/hr, if you ignore significant digits. If using significant digits, the answer would be limited to two significant digits and would then become 5.0 km/hr.
To calculate the direction, we would revisit that right triangle I mentioned earlier, and would use a trigonometric function to calculate the angle. I will calculate the measure of the top angle of the triangle. To do so, I am going to use tangent, remembering that the tangent of an angle equals the length of the side opposite the angle divided by the length of the side adjacent to it. Using θ to represent the angle, we get...
tanθ = 4.0 / 9.0
So, θ = arctan(4.0 / 9.0), which comes out to be 24 degrees east of south, if rounding to two significant digits.
By way of reminder, when you're finding arctan of a number, you are finding the angle that has that number as its tangent.
This leads us to the final answer of 5.0 km/hr 24 degrees east of south, for the average velocity.
