Chapter 10, Problem 82RE

a.

To determine

Find parametric equation that model the motion of the train and Mary as a function of time.

Answer to Problem 82RE

  $\boxed{\frac{3}{2}{t}^2,6\left(t-2\right)}$

Explanation of Solution

Given information:

Mary’s train leaves at $7:15AM$ and accelerates at the rate of $3meter/\sec ond$ . Mary who can run $6meters/\sec ond$ , arrives at the train station $2\sec onds$ after the train has left.

Find parametric equation that model the motion of the train and Mary as a function of time.

[hint: the position $s$ at time $t$ of an object having acceleration $a$ is $s=\frac{1}{2}a{t}^2$ .]

Calculation:

Let the position of train at time $t$ is given by $s_1=\frac{1}{2}a{t}^2$ where $a$ refers to acceleration of the train. $\begin{array}{l}{s}_1=\frac{1}{2}a{t}^2\\ =\frac{3}{2}{t}^2\end{array}$

The motion of Mary is given by equation,

  $\begin{array}{l}{s}_2=v\left(t-2\right)\\ =6\left(t-2\right)\end{array}$

Hence, the parametric equation for motion of train and Mary is $\boxed{\frac{3}{2}{t}^2,6\left(t-2\right)}$ .

b.

To determine

Determine algebraically whether Mary will catch the train. If so, when?

Answer to Problem 82RE

  $\boxed{\text{Never}}$

Explanation of Solution

Given information:

Mary’s train leaves at $7:15AM$ and accelerates at the rate of $3meter/\sec ond$ . Mary who can run $6meters/\sec ond$ , arrives at the train station $2\sec onds$ after the train has left.

Determine algebraically whether Mary will catch the train. If so, when?

Calculation:

Consider the parametric equation for motion of train and Mary,

  $\begin{array}{l}\frac{3}{2}{t}^2=6t-12\\ 3t^2=12t-24\\ t^2-4t+8=0\\ t=\frac{4\pm \sqrt{{\left(-4\right)}^2-4\left(1\right)\left(8\right)}}{2\times 1}\end{array}$

The time instant at which this will happens are imaginary .

Hence Mary will $\boxed{\text{Never}}$ catch the train.

c.

To determine

Simulate the motion of the train and Mary by simultaneously graphing the equations found in part $\left(a\right)$

Answer to Problem 82RE

  

Explanation of Solution

Given information:

Mary’s train leaves at $7:15AM$ and accelerates at the rate of $3meter/\sec ond$ . Mary who can run $6meters/\sec ond$ , arrives at the train station $2\sec onds$ after the train has left.

Simulate the motion of the train and Mary by simultaneously graphing the equations found in part $\left(a\right)$

Calculation:

Consider the parametric equation for motion of train and Mary, and graph the equation with respect to time.

  

From the graph the position of train and Mary are not same,

Hence, Mary will $\boxed{\text{Never}}$ catch the train.