Prelab 02: Computational Kinematics

Name: Hannah Black

Part One: Position Update Equation with Constant Velocity

Suppose a particle moves in the $-\hat{y}$ direction starting at $y=100\ \text{m}$ with a velocity $v_y = -10\ \text{m/s}$.

Prelab Problem 2.1: With the initial conditions $y=100\ \text{m}$ and $v_y = -10\ \text{m/s}$, how long will it take for the particle to reach $y=0$?

#Using the kinematic equation y(f) = y(0) + v(t(f) - t(0)) and solving for the change in t, delta t = (y(f) - y(0))/(v) = (0 - 100)/(-10) = 10 s

Let us confirm your answer to problem 2.1 numerically. To do so, we will use the position update equation $\vec{r}_\text{new} = \vec{r}_\text{old} + \vec{v}\cdot \Delta t$ which simply states that the position at a time $t_\textrm{new}$ is equal to the old position at time $t_\textrm{old}$ plus the change in position due to the velocity. This equation is essentially one step in the summation of an integral, we must perform it multiple times to do the entire integral. To to the full integral, we will use a while loop. In this case, we are starting at $y_0 = 100$ and taking time steps of $1 \ \text{s}$. We also have a constant velocity of $-10 \ \text{m/s}$ in the $\hat{y}$ direction.

Prelab Code Task 2.1: Input the initial conditions for the y position and y velocity. Then, run the cell to define the initial conditions.

Next we will set up the while loop to perform the integration.

Prelab Code Task 2.2: Read code below then run the cell below, without editing, to perform the integration.

Part Two: Characteristic Time

The characteristic time of a physical process is not fixed, but rather an estimate based on the physical properties of the problem. For example, if we consider the earth's orbit, it takes one year to complete an orbit therefore a good characteristic time would be 1 year. Another way to arrive at 1 year for an earth orbit would be to use the values of the problem's parameters in a combination that has units of time.

In a very similar manner, we can estimate the characteristic time for the process in part one under the influence of earth's gravity near the surface. Near the earth's surface $a = g = 9.81\ \text{m}/\text{s}^2$, and in the problem above $y_0 = 100\ \text{m}$.

Prelab Problem 2.2 Using the values of $g$ and $y_0$ above, estimate the characteristic time for an object falling $100\ \text{m}$ near the earth's surface where $g = 9.81\ \text{m}/\text{s}^2$. Hint: what combination of $g$ and $y_0$ has units of seconds?

square root of (y(0) (m) / (g (m/s^2)) gives units of only seconds, as the m^(1/2) cancels out and the s^2 becomes seconds

Part Three: Combining Velocity Update and Position Update Equations

Finally, we will include the influence of gravity on the object by using the acceleration due to gravity to update the velocity as the object falls. The velocity update equation is $\vec{v}_\text{new} = \vec{v}_\text{old} + \vec{a}\cdot \Delta t$. The acceleration due to gravity is $a_g = (0,-g,0)$ where $g$ is the magnitude of the gravitational acceleration. In this case the acceleration is simple and uniform, but in the future we will use the force with newton's second law to determine the acceleration for the velocity update, as $a = F/m$.

To include velocity updates, we will modify our previous code.

Prelab Code Task 2.3: Fill in the gaps in the code below to perform the computation of an object falling $100 \text{m}$ under the influence of gravity. Note: You will need the numpy square root function np.sqrt

You're done! Submit your pre-lab per the following instructions