Cheetahs have one of the highest accelerations of any land animal, allowing them to reach maximum speeds very quickly. Typically, a cheetah can accelerate from a standing start to about $$ 60 \mathrm{mi} / \mathrm{hr}(96 \mathrm{~km} / \mathrm{h}) $$ in less than 3 seconds. This translates to an acceleration rate of approximately $$ 20 \mathrm{~m} / \mathrm{s}^{2} $$. Assuming a constant acceleration of $$ 20 \mathrm{~m} / \mathrm{s}^{2}\left(\mathrm{a}=20 \mathrm{~m} / \mathrm{s}^{2}\right) $$, what equations of motion (kinematic equations) do you use to graph a position vs. time for a cheetah? Also assume the following: time interval of 3.0 s in intervals of 0.1 s initial time to $$ =0.0 \mathrm{~s} $$ initial position $$ x 0=0.0 \mathrm{~m} $$ initial velocity v0 $$ =0.0 \mathrm{~m} / \mathrm{s} $$ (cheetah initially at rest) Plot points in a scatter plot.

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For constant acceleration, the position of an object (in this case, a cheetah) is given by the kinematic equation:

x(t) = x₀ + v₀t + (1/2)at²

Using the values provided:
  • Initial position x₀ = 0.0 m
  • Initial velocity v₀ = 0.0 m/s
  • Acceleration a = 20 m/s²

This simplifies the equation to:

x(t) = 0.0 + 0.0·t + (1/2)(20)t² = 10t²

So, the position of the cheetah as a function of time is x(t) = 10t².

Additionally, the velocity at any time t is given by:

v(t) = v₀ + at = 0 + 20t = 20t

Since you are asked to graph a position vs. time plot, you will use the equation:

x(t) = 10t²

For a time interval of 3.0 s in increments of 0.1 s (i.e., t = 0.0, 0.1, 0.2, …, 3.0), you can calculate x(t) at each time step. Here is an example of how some of these calculated points would look:

Time (s)  Position (m)
  0.0      0.0     [10×(0.0)²]
  0.1      0.1     [10×(0.1)² = 10×0.01]
  0.2      0.4     [10×(0.2)² = 10×0.04]
  0.3      0.9     [10×(0.3)² = 10×0.09]
  0.4      1.6     [10×(0.4)² = 10×0.16]
  …      …      
  3.0      90.0     [10×(3.0)² = 10×9]

To create a scatter plot of these points (with time on the horizontal axis and position on the vertical), you can use your favorite plotting tool (such as Python’s matplotlib).

For example, in Python:

------------------------------------------------
import matplotlib.pyplot as plt
import numpy as np

# Create an array of time points from 0 to 3 seconds in 0.1 s intervals.
t = np.arange(0, 3.1, 0.1)
# Calculate positions using x(t) = 10t^2.
x = 10 * t**2

plt.scatter(t, x)
plt.title("Cheetah Position vs. Time")
plt.xlabel("Time (s)")
plt.ylabel("Position (m)")
plt.grid(True)
plt.show()
------------------------------------------------

This code will generate a scatter plot showing the cheetah’s position at each time interval.

In summary, the equation used for the graph is:

x(t) = 10t²

with t ranging from 0.0 s to 3.0 s in increments of 0.1 s.
To graph a cheetah's position over time with constant acceleration of 20 m/s², use the equation: x(t) = 10t² Plot points at 0.0, 0.1, 0.2, ..., 3.0 seconds to create a scatter plot showing position vs. time.

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