Question

Sam receives the kicked football on the 3 yd line and runs straight ahead toward the goal line before cutting to the right at the 15 yd line. He then runs 9 yd along the 15 yd line directly toward the right sideline before being tackled. What was Sam’s distance traveled? What was Sam’s resultant displacement? How many yards did Sam gain in this play (how far was the ball advanced toward the goal line)?

Answer 1

Answer:

Distance: 21 yd, displacement: 15 yd, gain in the play: 12 yd

Explanation:

The distance travelled by Sam is just the sum of the length of each part of Sam's motion, regardless of the direction. Initially, Sam run from the 3 yd line to the 15 yd line, so (15-3)=12 yd. Then, he run also 9 yd to the right. Therefore, the total distance is

d = 12 + 9 = 21 yd

The displacement instead is a vector connecting the starting point with the final point of the motion. Sam run first 12 yd straight ahead and then 9 yd to the right; these two motions are perpendicular to each other, so we can find the displacement simply by using Pythagorean's theorem:

$d=\sqrt{12^2+9^2}=15 yd$

Finally, the yards gained by Sam in the play are simply given by the distance covered along the forward-backward direction only. Since Sam only run from the 3 yd line to the 15 yd line along this direction, then the gain in this play was

d = 15 - 3 = 12 yd