Question

TACTICS BOX 1.5 Drawing a pictorial representation

Draw a motion diagram. The motion diagram develops your intuition for the motion and, especially important, determines whether the signs of v?  and a?  are positive or negative.

Establish a coordinate system. Select your axes and origin to match the motion. For one-dimensional motion, you want either the x axis or the y axis parallel to the motion.

Sketch the situation. Not just any sketch. Show the object at the beginning of the motion, at the end, and at any point where the character of the motion changes. Show the object, not just a dot, but very simple drawings are adequate.

Define symbols. Use the sketch to define symbols representing quantities such as position, velocity, acceleration, and time. Every variable used later in the mathematical solution should be defined on the sketch. Some will have known values, and others are initially unknown, but all should be given symbolic names.

List known information. Make a table of the quantities whose values you can determine from the problem statement or that can be found quickly with simple geometry or unit conversions. Some quantities are implied by the problem, rather than explicitly given. Others are determined by your choice of coordinate system.

Identify the desired unknowns. What quantity or quantities will allow you to answer the question? These should have been defined as symbols in step 4. Don’t list every unknown, only the one or two needed to answer the question.

Follow the steps above to draw a pictorial representation of the following problem: A light train is traveling on a straight section of track at a constant speed of 15 m/s. As it approaches the next station, it starts to slow down at a rate of 5 m/s2 until it stops at the station. From the moment the train starts to slow down, how long does it take for the train to reach the station? Note that you are not expected to solve this problem, only to draw the pictorial representation.

Part A

Draw a motion diagram for the train. Assume that the train is moving toward the right and starts to slow down at t=0. The separation of each dot represents an elapsed time of one second. Include velocity vectors and the acceleration vector in your drawing. Keep in mind that the acceleration of the train has a magnitude of 5 m/s2.

Part B

Draw the coordinate system that is most appropriate for the train's motion. Since this problem deals with one-dimensional motion, you need only the x-axis. Note that xi, xf, ti, tf, (vx)i and (vx)fare the initial position, final position, initial time, final time, initial velocity, and final velocity of the train, respectively.

Keep in mind that the choice of coordinate system is arbitrary. However, in problems involving one-dimensional motion it is often convenient to select the x-axis to match the motion. Thus, in this case the positive end of the x-axis is chosen to be to the right.

Though it is an arbitrary choice, it is often convenient to place the origin at the location where the nature of the motion changes. In this problem, it is where the train starts to slow down. We will assume this for Part C.

Part C

To complete your pictorial representation of the problem, you should compile two lists: one of known quantities and one of the unknown quantities that will allow you to answer the question in the problem. Below are all of the relevant quantities in this problem. Sort them accordingly.

Drag the appropriate items to their respective bins.

I only need Part C

Answer 1

Concepts and reason

The main concept required to solve this question is the kinematic equation of motion.

Initially, calculate the total time taken by the train to reach the station. Later, draw the motion diagram for the train and finally draw the coordinate system that is most appropriate for the train’s motion. Also, list down the known quantities and the unknown quantities that are used, while draw the diagram.

Fundamentals

The kinematic first equation of motion is,

$v = u + at$

Here, v is the final velocity, u is the initial velocity, a is the acceleration of the train, and t is the time taken by the train.

The second equation of motion is

$s = ut + \frac{1}{2}a{t^2}$

Here, s is the distance covered by the train, u is the initial velocity, and a is the acceleration of train.

The third equation of motion is

${v^2} = {u^2} + 2as$

Here, v is the final velocity, u is the initial velocity, a is the acceleration and s is the distance covered by the train.

The time taken by the train to reach the station is calculated as follows:

$v = u + at$

Here, v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time taken by the train to reach the station.

Substitute 0 m/s for v, 15 m/s for u and $- 5{\rm{ m/}}{{\rm{s}}^{\rm{2}}}$ for a,

$\begin{array}{c}\\v = u + at\\\\0{\rm{ m/s = 15 m/s + }}\left( { - 5{\rm{ m/}}{{\rm{s}}^{\rm{2}}}} \right)t\\\\t = 3{\rm{ s}}\\\end{array}$

a)

The motion diagram of train at various instant of time is drawn as follows:

b)

The coo-ordinate system is drawn as follows by taking time along x axis,

(c)

The known quantities are as per the kinematic equation of motion are,

$\begin{array}{l}\\{x_{\rm{i}}} = 0{\rm{ m,}}\\\\{{\rm{a}}_{\rm{x}}} = - 5{\rm{ m/}}{{\rm{s}}^{\rm{2}}}\\\\{\left( {{v_x}} \right)_{\rm{i}}} = 15{\rm{ m/s}}\\\\{\left( {{v_x}} \right)_{\rm{f}}} = 0{\rm{ m/s}}\\\\{t_{\rm{i}}} = 0{\rm{ s}}\\\end{array}$

Here, x_i is the initial distance, a_x is the acceleration along x axis, (v_x)_i is the initial velocity along x axis and t_i is the initial time taken by the train.

The unknown quantities are

$\begin{array}{l}\\{x_{\rm{f}}}\\\\{t_{\rm{f}}}\\\end{array}$

Here, x_f is the final distance covered by the train and t_f is the final time.

Ans: Part a

The motion diagram for the train reaching the station is .

Part b

The Cartesian representation diagram is:

The known quantities are:

$\begin{array}{l}\\{x_{\rm{i}}} = 0{\rm{ m,}}\\\\{{\rm{a}}_{\rm{x}}} = - 5{\rm{ m/}}{{\rm{s}}^{\rm{2}}}\\\\{\left( {{v_x}} \right)_{\rm{i}}} = 15{\rm{ m/s}}\\\\{\left( {{v_x}} \right)_{\rm{f}}} = 0{\rm{ m/s}}\\\\{t_{\rm{i}}} = 0{\rm{ s}}\\\end{array}$ and

The unknown quantities are $\begin{array}{l}\\{x_{\rm{f}}}\\\\{t_{\rm{f}}}\\\end{array}$ .