1-D kinematics

Say whether the following statements are true or false:

a) Reference frames can be classified as inertial and heterogeneous.
b) The study of motion is independent of the observer´s position in relation to the motion.

Find the position vector and its magnitude for the following points:

a) P₁ (-1,2)
b) P₂ (-3,4)

The S.H.S. Madrid-Barcelona during the first stage of its trip travels east at a rate of 50 m each second and North at 30 m each second. Calculate:

1. The motion x-coordinate as function of time
2. The motion y-coordinate as a function of time
3. The position equation of the train for the studied stretch
4. The distance to the origin of the section as a function of time

A body moves between any two instants of time following a circular path with a radius of 5 meter, as you can see in the figure.

Determine:

1. The displacement vector of the body and the distance traveled, assuming the origin of the system of reference is located at the starting point of the motion
2. The displacement vector of the body and the distance traveled, assuming the origin of the system of reference is in the center of the semicircle

Answer if the following statements are true or false:

a) Distance traveled only depends on the starting and final positions without consideration of the trajectory.
b) Distance traveled is always greater or equal than the magnitude of the displacement vector.
c) Distance traveled it is a vector.
d) As time passes, a body in motion always increases its displacement.

If a body is in the position (1,2) and after 2 seconds it is in the position (1,-2).

What will be its average velocity considering that all units are expressed in the International System?

Knowing that the distance a body travels as a function of time is given by the following equation:

$$S\left(t\right)=t^2+5·t+1$$

Calculate:

a) The average speed during the first 3 seconds.
b) The instantaneous speed of the body.

Given the following graphs that show the variation of the speed with time of three different bodies, answer the following questions:

a) Which case has the fastest variation of the speed?
b) What does the slope represent in each one of the 3 cases?
c) Why is the line in case I horizontal?
d) In case I, does the possibility exist that the body is experiencing acceleration?

A basketball player throws the ball with a velocity of ${\overrightarrow{v}}_1=-2·\overrightarrow{i}+\overrightarrow{j} m/s$, with such bad luck that it bounces off the board with a velocity ${\overrightarrow{v}}_2=15·\overrightarrow{i}+3·\overrightarrow{j} m/s$.

Calculate the average acceleration knowing that the impact against the board lasts exactly 0.02 seconds

A meteorite moves through space with a velocity v⃗(t) = (1+4·t) i⃗+t² j⃗ m. Calculate

a) Its average acceleration between the times t₁=2 s and t₂= 4 s.
b) Its acceleration at t₃= 6 s.

Knowing that the magnitude of the velocity of a body in S.I. units is:

$$v=7+2·t+3·t^2$$

Calculate the magnitude of the tangential acceleration.

Answer if the following statements are true or false:

a) If motion has normal acceleration, the motion is rectilinear.
b) If motion has tangential acceleration, the motion is curvilinear.
c) Normal and tangential acceleration are called intrinsic components of the acceleration.
d) Motion without normal acceleration and with constant tangential acceleration is called uniformly accelerated rectilinear motion. (u.a.r.m.)

Assuming that the following vector magnitudes refer to rectilinear motion, give their corresponding scalar representation:

• $\overrightarrow{r}=3·\overrightarrow{i} m$ 
• $∆\overrightarrow{r}=-3·\overrightarrow{j}m$ 
• $\overrightarrow{v}=4·(-\overrightarrow{j}) m/s$ 
• $\overrightarrow{v}=3·t·\overrightarrow{i}m/s$ 

A cyclist starts his morning ride and after 10 seconds his velocity is 7.2 km/h. At that moment, he sees a dog approaching and slows down for 6 seconds until the bicycle stops. Calculate:

a) The acceleration until he begins to slow down.
b) The braking acceleration of the bike.
c) The total distance traveled.