Position, path and displacement

You want to explain over the phone to a friend where your backpack is in the classroom. What point in the classroom would you choose as a reference to describe its position? Explain in your own words what a frame of reference is.

Along a straight line measured in meters, Mateo is at position $x = 3\ \text{m}$, taking a tree as the origin, as shown in the figure.

What does this data mean? Where would Mateo be if his position were $x = -3\ \text{m}$?

Leo says: "I am $10\ \text{m}$ away." His sister asks him: "$10\ \text{m}$ away from where?"

1. a) Explain what information is missing from Leo's sentence and why a position always requires a frame of reference.
2. b) Leo is $10\ \text{m}$ from a lamppost, and the lamppost is $6\ \text{m}$ from the bus stop, all on the same straight street. How far from the bus stop could Leo be? Give both possibilities.

An ant walks along the edge of a square table until it returns to the starting point. Is its path straight or curved? Explain what the path of a motion is.

A kitchen robot moves forward $3\ \text{m}$ in a straight line, turns at a right angle, and moves forward another $4\ \text{m}$, as shown in the figure.

What total distance has it covered?

You are sitting on a train passing through a station at $30\ \text{m/s}$.

1. a) Are you moving relative to your seat?
2. b) Are you moving relative to the station platform?
3. c) Using this example, explain what the sentence "motion is relative" means.

Sophia walks along a straight corridor: she moves forward $4\ \text{m}$ to the right and then moves back $3\ \text{m}$ to the left, as shown in the figure.

1. a) What total distance has she covered?
2. b) What was her displacement?

Can a moving object have a displacement equal to zero and yet have covered a distance greater than zero? Give reasons for your answer and provide an example.

Sara leaves the front door of her house (the origin), walks $60\ \text{m}$ in a straight line toward the park, and then turns back $25\ \text{m}$ toward home, as shown in the figure.

1. a) What distance has she covered?
2. b) What is her final position?
3. c) What was her displacement?

A robot vacuum cleaner moves along a straight hallway: it moves forward $5\ \text{m}$, moves back $3\ \text{m}$, and finally moves forward $2\ \text{m}$, as shown in the figure.

1. a) What total distance has it covered?
2. b) What was its displacement?
3. c) Do both values coincide? Explain in what type of motion distance and displacement are equal.

Mateo goes from his house to the newsstand, which is $200\ \text{m}$ away, and returns home by the same route.

1. a) What distance has he covered?
2. b) What was his total displacement?

An airplane flies from Chicago to Denver. Explain, with a diagram drawn in your notebook, the difference between the airplane's path and its displacement.

Mateo is at position $x = 2\ \text{m}$ in a straight hallway and moves to position $x = 9\ \text{m}$, as shown in the figure.

1. a) What was his displacement?
2. b) Has he necessarily traveled $7\ \text{m}$? Explain the difference between distance traveled and displacement.

You throw a ball upward: it goes up $2\ \text{m}$ and returns to your hand, as shown in the figure.

1. a) What distance has the ball covered?
2. b) What was its displacement?
3. c) Repeat both calculations if you catch the ball when it has only fallen $1\ \text{m}$.

An elevator starts from the basement, at position $x = -3\ \text{m}$, and goes up to the floor at $x = 5\ \text{m}$.

1. a) What was its displacement?
2. b) If it then goes down to position $x = 0\ \text{m}$, what total distance has it covered since leaving the basement?

A mine elevator starts from position $x = 10\ \text{m}$, goes down to $x = -90\ \text{m}$, and then goes up to $x = -30\ \text{m}$, as shown in the figure.

1. a) What total distance has it covered?
2. b) What was its total displacement?
3. c) Calculate the displacement for each stage separately and check that their sum equals the total displacement.

To go from home to school, Emma can take a straight street of $600\ \text{m}$ or go around the park, which is $850\ \text{m}$, as shown in the figure.

1. a) How are the distance traveled and displacement similar, and how do they differ for each route?
2. b) Can displacement be greater than the distance traveled? Justify your answer.

A remote-controlled car starts at position $x = 2\ \text{m}$ and makes the following successive movements: $+8\ \text{m}$, $-5\ \text{m}$ and $-3\ \text{m}$.

1. a) What is its final position?
2. b) What distance has it covered?
3. c) What is its total displacement? What is notable about this result?

A cyclist trains in a circular velodrome where each lap is $250\ \text{m}$, as shown in the figure.

1. a) After completing $4$ full laps, what distance has he covered and what is his displacement?
2. b) What if he completes $4$ and a half laps? Calculate the distance and explain why the displacement is no longer zero: what feature of the track is it equal to?

An athlete completes one full lap of a $400\ \text{m}$ track, as shown in the figure.

1. a) What distance has she covered?
2. b) What was her displacement?
3. c) Answer the same questions if she completes two full laps and explain why the displacement is zero even though the distance is not.