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, Pablo 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 Pablo 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?

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

Lucía 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?

Hugo goes from his house to the kiosk, 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?

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.

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

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

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.

Sara leaves the front door of her house (the origin), walks $60\ \text{m}$ in a straight line towards the park, and then turns back $25\ \text{m}$ towards 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?

You throw a ball upwards: 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 $1\ \text{m}$.

A lift 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 robot vacuum cleaner moves along a straight corridor: 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.

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?

A mine lift 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, Vega 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 covered and displacement similar, and how do they differ for each route?
2. b) Can displacement be greater than the distance covered? 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?

Compare these two motions: a coin falls in a straight line $1\ \text{m}$ to the ground, and a ball, following a curved path, travels $10\ \text{m}$ and ends up $1\ \text{m}$ lower and $3\ \text{m}$ horizontally from where it was thrown.

1. a) In which of the two do the distance travelled and the displacement coincide?
2. b) In which one do they not coincide? What is the value of the displacement in that case?
3. c) State a general rule: when do distance and displacement coincide?

A drone flies $30\ \text{m}$ north and then $40\ \text{m}$ east, as shown in the figure.

1. a) What distance has it covered?
2. b) Calculate its displacement. Hint: the displacement is the hypotenuse of a right-angled triangle; use Pythagoras' theorem.

A postman walks around a block: he walks $60\ \text{m}$ along one street, turns at a right angle, and walks $80\ \text{m}$ along the next street.

1. a) What distance has he covered?
2. b) If he could return to the starting point in a straight line, how far would he walk? Use Pythagoras' theorem.

A rectangular pitch measures $30\ \text{m}$ wide by $40\ \text{m}$ long. One player goes from one corner to the opposite corner along the edge of the pitch, and another crosses along the diagonal, as shown in the figure.

1. a) What distance does each person cover?
2. b) What is the displacement of each person?
3. c) What conclusion do you draw about distance and displacement?

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.