Related questions with answers

Given two points on a line and a third point not on the line, is it possible to draw a plane that includes the line and the third point. Explain your reasoning.

You and your friend walk in opposite directions, forming opposite rays. You were originally on the corner of Apple Avenue and Cherry Court. a. Name two possibilities of the road and direction you and your friend may have traveled. b. Your friend claims he went north on Cherry Court, and you went east on Apple Avenue. Make an argument as to why you know this could not have happened.

One can define the collision time $\tau$ in the following way: The probability $P$ that an electron suffers a collision during an infinitesimal time interval $d t$ is

P(\text { collision in time } d t)=d t / \tau

Note that the probability that the electron suffers no collision during a time interval $d t$ is then given by

P(\text { no collision in } d t)=1-d t / \tau

Consider now the probability $P^{\prime}(t)$ that a given electron undergoes no collision during a finite time interval $t$ . (a) Show that $P^{\prime}$ obeys the equation

\frac{d P^{\prime}}{P^{\prime}}=-\frac{d t}{\tau}

[Hint: First, argue that $P^{\prime}(t+d t)=P^{\prime}(t)(1-d t / \tau)$ .] (b) Using the above equation, solve for $P^{\prime}(t)$ and sketch this function. (c) Suppose that a collision occurs at time $t=0$ . Argue that the probability $p^n(t) d t$ that the next collision occurs between time $t$ and $t+d t$ is given by

p^{\prime \prime}(t) d t=e^{-t / \tau} \frac{d t}{\tau}

(d) Use the above equation, to compute the average time, $\langle t\rangle$ , between collisions. (e) Compute the rms average time $\sqrt{\left\langle t^2\right\rangle}$ between collisions.