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# 1 Questions

## 1.1 Wheels of a moving car

You are on the side of the street looking at a cars passing by. A car is passing by. At time \(t=0\) the center of the front wheel of the car is exactly in front of you and you notice that it is travelling at a speed of \(44 \text{ feet}/\text{sec}\). You need to describe to someone the motion of the center of the front wheel of the car as time passes.

### 1.1.1

You decide to use a function.

- What is the input variable of the function. What physical quantity does it represent?
- What is (what kind of an object e.g. real number, vector?) the output of the function if you only care about describing how far down the road the center of the wheel is travels?
- What is (what kind of an object e.g. real number, vector?) the output of the function if you care also how high above the road level the center of the wheel is?
- Write out the function that describe the two cases above. The diameter of the wheel is \(1.6 \text{ feet}\).

### 1.1.2

You notice that at time \(t=0\) the car wheel hits a piece of gum on the road surface that sticks to the wheel.

The formula that relates the speed of rotation to the speed of the car is \(\text{rotation speed in revolutions/sec}=\frac{\text{velocity of car}}{2 \pi \text{ (radius of wheel)}}\) giving \(\text{(rotation speed)}\approx 8.75 \text{ rev}/\text{sec}\). If you have time, think how one can obtain this formula.

- From the point of view of the car, describe (using a function) the motion of the stuck piece of gum relative to the center of the front wheel.
- From your point of view (on the sidewalk) describe (using a function) motion of the stuck piece of gum relative to the center of the front wheel.
- At every moment in time describe how quickly is the piece of gum travelling along the road.
- At every moment in time describe how quickly is the piece of gum travelling up and down.
- What would you consider the "velocity vector" of the piece of gum?

### 1.1.3

- Suppose that the car slows down a bit on the way (but does not skid). What answers to the above questions would change?
- Would the path travelled by the piece of gum change?

## 1.2 Dimensions and multiple coordinates:

- Give an example of a physical quantity that is described by a real number.
- Give an example of a physical quantity that is described by a 2D vector.
- Give an example of a physical quantity that is described by a 3D vector.
- Give an example of a physical process described by a functions from real numbers to real number.
- Give an example of a physical process described by a functions from real numbers to 2D vectors.
- Give an example of a process described by a functions from real numbers to 3D vectors.
**hard**Can you come up with an example of a physical quantity that is described by a 4D or higher-D vector?

## 1.3 Rotations

You want to describe to someone the position of the steering wheel: how much it is turned.

### 1.3.1

One way of doing it is putting a pin at the topmost position of the steering wheel when it is "straight" and then giving the coordinates of with respect to the center of the wheel of the pin.

- Do you think this approach works?
- If the radius of the steering wheel is 1 unit, what is the description according to the method above for "turned 45 deg to the left"?
- If the radius of the steering wheel is 1 unit, can the vector \([-1;2]\) describe a valid position of the steering wheel according to the method above?

### 1.3.2

We could also describe the position just by giving an angle: \(0\) is the "staight" position. \(+\pi\) is half turn to the right \(-\pi\) is half turn to the left.

- Do you think this approach works?
- Do different angles always correspond to different positions of the steering wheel?
- What if it was not a steering wheel but a round pizza positioned in the middle of a table (you don't care how many complete rotations you did)?

### 1.3.3

Which do you think is more correct:

- The position of the steering wheel is correctly described by a 1D quantity (number).
- The position of the steering wheel is correctly described by a 2D quantity (vector).

## 1.4 Rotations (advanced)

**very hard/optional** (you are welcome to discuss on piazza or during office hours)

- You want to describe the position of a soccer ball in space. First you give the \([x,y,z]\) vector representing the center of the ball in space. But now you also need to describe the way it is rotated. How do you do it?
- How many coordinates do you need to describe the rotation of the ball with center at \([0,0,0]\)?
- What is the dimension of this physical quantity?