Rectilinear Motion I

  • Topics Covered :

Kinematics- Basic concepts, Variable acceleration, Motion curves, Relative motion

  • Motion

By the word motion, we mean a change in position of an object with respect to some fixed location as well as with respect to time. We observe motion at various length and time scales. For example, fingers of a pianist move w.r.t. his body (hand), while traveling, we move ourselves w.r.t. the road, planets rotate about the sun, an electron goes around the nucleus. Another example would be you walking in a moving train.

We note 2 important points. First, the presence of a location, w.r.t. which the movement is specified. The human body, the earth, the sun, the nucleus and the train are all references. These references are themselves fixed or they may move (the train for example). These are known as frames of reference. Second, the positional change is measured with a time frame (clock), associated with the frame of reference. For example, the speedometer of the car tells you how fast you’re moving w.r.t. ground.

Thus, a frame of reference is required (or assumed without specifying) to make sense out of a motion. Generally, the earth is the fixed frame of reference for all mechanical engineering related applications.

  • Types of Motion

Primarily, rectilinear and curvilinear motions are 2 types. A sprinter running in a 100m race is under a rectilinear motion, since he runs in a straight line. A missile launched in space is under curvilinear (projectile) motion. What about the motion of wheels of a bicycle?

  • Rectilinear Motion

It is the motion in a straight line, and is the easiest to analyze. Few important terms are:

Distance (s) is the length of path traveled by a particle. It is a non-negative scalar quantity.

Displacement (\vec r) is the shortest distance between the initial and the final position of a body.

Position Vector (\vec r_p) gives the location of the body w.r.t. a reference point.

Average Speed (v_{avg}) is the total distance traveled divided by total time taken.

Instantaneous Speed (v_{inst}) is the time derivative of distance. Hence,

{v_{avg} = \frac {total \ distance (s)}{total \ time (t) } \ , \ v_{inst} = \frac {ds}{dt}}

Average Velocity (\vec v_{avg}) is the total displacement divided by total time.

Instantaneous Velocity (\vec v_{inst}) is the time derivative of displacement.

\vec v_{avg} = \frac {\vec r}{t}, \vec v_{inst} = \frac {d \vec r}{dt}

Instantaneous Acceleration \vec {a}_{inst} is the time derivative of instantaneous velocity. Thus,

\vec a_{inst} = \frac {d \vec v_{inst}}{dt}

Jerk \vec j is the time derivative of instantaneous acceleration, i.e.

\vec j = \frac {d \vec a_{inst}}{dt}

Note: Unless otherwise specified, we will be dealing with the instantaneous quantities. These terms are important in studying curvilinear motion as well.

  • Few Things from Calculus

If y is a function of x, then

1) slope of tangent to the curve y=f(x) is given by \frac {dy}{dx} and

2) the area under the curve from x=a to x=b is given by \int \limits_a^b f(x)dx

Integration of a polynomial function increases the degree by 1 and differentiation of a polynomials function decreases the degree by 1.

We will now apply the knowledge of calculus to analysis of rectilinear motion.

  • Motion Diagrams

Since a = \frac {dv}{dt} and v = \frac {dr}{dt}, we can say that

1) slope of v-t diagram gives the acceleration,

2) the area under v-t diagram from t=t_1 to t=t_2 gives the displacement.

Since j= \frac {da}{dt} and a = \frac {dv}{dt}, we can say that

1) slope of a-t diagram gives the jerk and

2) the area under a-t diagram from t=t_1 to t=t_2 gives the change in velocity.

Since v = \frac {dr}{dt}, we can say that slope of r-t diagram gives the velocity.

Recall, a = \frac {dv}{dt} = \frac {dv}{dr} \times \frac {dr}{dt}. This is chain rule. So,

a = \frac {dv}{dr} \times v = v \frac {dv}{dr}

So, from v-r diagram, we can get the acceleration as v \frac {dv}{dr}

  • Examples

Motion under gravity (free fall) is an example of rectilinear motion. Another example is motion of elevator. Car moving on a straight road could be another example.

  • Relative Motion

Suppose you are sitting in a moving train, which is moving with sufficiently large speed (say 60 \ kmph). You cross another train, which is moving in opposite direction with some speed (say 50 \ kmph). You would notice that the other train appears to travel faster than it actually is. This is an example of relative motion.

Now, if another train moves with same speed (50 \ kmph) in same direction as you are, it will appear to move slower than it actually moves. This is another example of relative motion.

The apparent values are 60+50 = 110 \ kmph and 60-50 = 10 \ kmph respectively.

Thus, when 2 objects move relative to each other, the motion of one with respect to other is a relative motion. Let \vec v_A and \vec v_B be the velocities of 2 objects A and B. To make things simpler, we imagine that one of the objects is stationary (say A). This is done by adding a vector - \vec v_A to both the motions.

relative

Thus, from a perspective of object A, B is moving with a velocity of \vec v_B - \vec v_A. Hence relative velocity of B with respect to A is

\vec v_{relative} = \vec v_{B/A} = \vec v_B - \vec v_A

Note that \vec v_{A/B} = - \vec v_{B/A}.

The relative acceleration \vec a_{B/A} is given by

\vec a_{B/A} = \vec a_{B} - \vec a_{A}

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