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NO ME SALEN
PROBLEMAS RESUELTOS DE FÍSICA DEL CBC
(Vectorial kinematics) 		 

manolito

NMS 1.2- Diego´s house is located 15 blocks (1,5km ) to the East and 20 blocks (2km) to the South, from Silvia´s house. One day they decide to go together to the cinema; the movie starts at 19:30h, and the cinema is located 30blocks to the East and 40 blocks to the North of Diego´s house.
Diego picks up Silvia: to accomplish this, he must catch the bus, wich will take him from his house to Silvia´s, with an average speed vector :
        vm = — 6 km/h i + 8 km/h j (i to the East; j to the North).
He has to wait 10 minutes for Silvia to finish with her make-up, then, they have to ride a cab, wich will drop them at the cinema 15 minutes after.

a- Choose a reference system, and determine the position vector of both houses and the cinema.

b- Find the displacement vectors and the taxi´s average speed.

c- If they reached the cinema 5 minutes after the movie begun, calculate the time when Diego leaves his house.

d- Determine the average speed vector of Diego and his modulus, in the trip between his house and the cinema.
This problem is a tsar bomba in terms of atomical silliness. Most of the time in this problem, I spent it doing the scheme. Look at it very carefully, I will guide you.    
Gráfico - No me salen - Ricardo Cabrera

Fisrt thing you wanna do, is checking the position of the three meeting points: D (Diego´s house), S (Silvia´s house) and C (cinema), check that the relative positions in the scheme match the ones from the statement above.

I chose the center coordinate in an arbitrary spot, any other spot would do the trick anyway.

The grid represents 5 blocks (half a kilometer) for each square.
Si no te acordás cómo se suman y restan vectores, podés ir a mis apuntes sobre vectores.

a - According to the reference system, the position vectors (the red ones) are:

        rD = 1 km î — 3 km ĵ

        rS = — 0,5 km î — 1 km ĵ

        rC = 4 km î + 1 km ĵ

î and ĵ are vectors of modulus 1, also called ``Versors´´, their objetive is to join the vectorial characteristics of a number and tell where it is pointing to. Usually it is portrayed with a little hat or a little comma above ( replacing the typical i and j ) . Versor î has the same direcction and sense in the x axis, as the ĵ versor in the y axis.

b - The displacement vector, ΔrDS (the bus from Diego´s house to Silvia´s) is the vectorial subtraction between the rear position vector and it´s previous one, rS — rD; To find it, you must subtract component by component. It´s easy, just look.

ΔrDS =  rS — rD

ΔrDS = — 1,5 km î + 2 km ĵ

This is the green one, with it´s origin in Diego´s house and it´s end in Silvia´s house, it does not mean that the bus went from one point to the other moving on a straight line as it was an helicopter. It´s about displacement, not trajectory.

Those displacement vectors are independent from the reference system you have chosen. If you want you can try with other reference system, you will get the same vector.

ΔrSC =  rC — rS

ΔrSC = 4,5 km î + 2 km ĵ

Esos vectores desplazamiento son independientes del SR elegido. Probá vos con otro SR... vas a ver que te dan lo mismo.

c - Las velocidades medias son los cocientes entre los desplazamientos y los intervalos de tiempo insumidos en cada desplazamiento. Eso es lo que nos va a permitir saber a qué hora salió Diego de su casa.

vmDS = ΔrDS / ΔtDS

That average speed is a fact that you have in the statement above... then:

— 6 km î + 8 km ĵ = (— 1,5 km î + 2 km ĵ ) / ΔtDS

The division is also applied component by component, so...

— 6 km î = — 1,5 km î / ΔtDS

8 km ĵ = 2 km ĵ / ΔtDS

Logically, both calculations should give the same answer... and they do: the time spent in the bus is 0.25 h, that is 15 minutes.

That means: Diego left his house at 18:55. The movie started at 19:30... but they arrived at 19:35 (5 minutes late). They left Silvia´s house fifteen minutes before the arrival, that is 19:20. But Diego arrived at Silvia´s house 10 minutes earlies, at 19:10, and he waited will she was on make up. Women! if he took 15minutes to arrive at her house, that means he left his house at 18:55.

d - to calculate Diego´s average speed vector from his house to the cinema, fisrt we must finf his displacement... (wich I didn´t represent in the scheme) and after that, we divide it by the total time interval:

ΔrDC =  rC — rD

ΔrDC = 3 km î + 4 km ĵ

The time interval between Diego´s departure and his arrival to the cinema is equal to 40 minutes, that is 0,66 h

vmDC = ΔrDS / ΔtDS

vmDC = ( 3 km î + 4 km ĵ ) / 0,66 h

vmDC = 4,5 km/h î + 6 km/h ĵ

To find the modulus of that vector, we apply Pitagoras:

|vmDC| = [ (4,5km/h)² + (6 km/h )² ]½

|vmDC| = 7,5 km/h

(not bad for a taxi)

  
   
 
CHALLENGE: redo the excercise with a reference system orientated the same way as mine, but with it´s origin located 5km to the North and 4km to the West from D   Ricardo Cabrera
 
Some rights reserved. Reproduction allowe by quoting the source. Translated by Santi Pisci. Last Updated jul-15. Buenos Aires, Argentina.