Question

1. You’re riding a unicorn at 25 m/s and come to a uniform stop at a red light 20 m away. What’s your acceleration?

Answer 1

Task:

1. You're riding a unicorn at $25 \, \text{m/s}$ and come to a uniform stop at a red light $20 \, \text{m}$ away. What's your acceleration?

Solution:

Due to Newton's second law:

$$\sum F = m a$$ $$a = \frac {\sum F}{m}$$ $$a = \dot {v}$$ $$v = \int a d t = a t + v _ {0}$$ $$t = \left(\frac {v - v _ {0}}{a}\right)$$ $$v = \dot {s}$$ $$\begin{array}{l} s = \int v d t = \int (a t + v _ {0}) d t = \frac {a t ^ {2}}{2} + v _ {0} t + s _ {0} = \frac {a \left(\frac {v - v _ {0}}{a}\right) ^ {2}}{2} + v _ {0} \left(\frac {v - v _ {0}}{a}\right) + s _ {0} = \\ = \frac {v ^ {2} - 2 v v _ {0} + v _ {0} ^ {2}}{2 a} + \frac {v v _ {0} - v _ {0} ^ {2}}{a} + s _ {0} = \frac {v ^ {2} - v _ {0} ^ {2}}{2 a} + s _ {0} \\ \end{array}$$

Given:

$$s = 2 0 m,$$ $$v _ {0} = 2 5 \frac {m}{s}$$ $$s _ {0} = 0 m$$ $$v = 0 \frac {m}{s}$$ $$2 0 m = - \frac {\left(2 5 \frac {m}{s}\right) ^ {2}}{2 a}$$ $$a = - \frac {\left(2 5 \frac {m}{s}\right) ^ {2}}{2 0 m} = - 3 1. 2 5 \frac {m}{s ^ {2}}$$

Answer:

$$a = - 3 1. 2 5 \frac {m}{s ^ {2}}$$