chapter 2 | motion along a straight line

Position and DisplacementAverage Velocity and Average Speed

Instantaneous Velocity and SpeedAcceleration

1D Motion with Constant AccelerationFree-Fall Acceleration

CHAPTER 2 | MOTION ALONG A STRAIGHTLINE

H. Oymak

Physics GroupAtilim University

February 2013

H. Oymak CHAPTER 2 | MOTION ALONG A STRAIGHT LINE




Outline

1 Position and Displacement

2 Average Velocity and Average Speed

3 Instantaneous Velocity and Speed

4 Acceleration

5 1D Motion with Constant Acceleration

6 Free-Fall Acceleration





Introduction

Kinematics: the branch of physics describing the motionof bodies without the causes of the motion.

One-dimensional (1D) motion: the motion of bodiesalong a straight line.

Objects in this chapter are to be considered as particles:(1) point-like or (2) rigid body.





Position

The position of an object moving along the x-axis will bespecified by its x-coordinate with respect to the origin.

x

negativedirection

positivedirection

origin

0 1 2 3−1−2−3





Displacement

The displacement ∆x of an object is the change in itsposition:

∆x = x2 − x1 (m)

where x1 its position at time t1, and x2 at a later time t2.





Position vs Time Graph

The time-dependent position x(t) of an object will bepictured as a position vs time curve:

x

t0





Average Velocity

The average velocity vavg gives a rough estimate of “howfast” an object undergoes a displacement over a timeinterval:

vavg =∆x∆t

=x2 − x1

t2 − t1(m/s)





Average velocity is equal to the slope of the line passingthrough points (t1, x1) and (t2, x2), so that

vavg =∆x∆t

=x2 − x1

t2 − t1= tan θ

x

t

∆

∆x

tθ

t1 t2

x1

x2





Average Speed

Average speed savg is the ratio of the total distancecovered by an object to the time interval ∆t :

savg =total distance

∆t(m/s)





(Instantaneous) Velocity

The (instantaneous) velocity v(t) of an object gives theinformation about how fast it is at any given instant t . It isthe time-derivative of position x(t):

v(t) =dxdt

(m/s)





v(t) is equal to the slope of x-t curve at any given instant:

moves left

stops

moves right

t

x





Speed

The magnitude of velocity is called speed; it is always apositive quantity.





Average Acceleration

The rate at which the velocity of an object changes is itsacceleration.

If an object have velocity v1 at time t1 and velocity v2 at alater time t2, its average acceleration aavg is given as

aavg =∆v∆t

=v2 − v1

t2 − t1

(

m/s2)





(Instantaneous) Acceleration

The (instantaneous) acceleration of an object at anygiven time is

a(t) =dvdt

(

m/s2)

The slope of v(t) at any point gives the acceleration of theobject at that point.





Acceleration is also the second derivative of displacement:

a(t) =dvdt

=ddt

(

dxdt

)

=d2xdt2





v

a

x

t

t

t

Figure: Successive graphs of acceleration, velocity, and displacement for an object.





1D Motion with Constant Acceleration

a

t

t

t

x

v

0v

x0

Figure: Representative graphs of a(t), v(t), and x(t) for an object moving withconstant acceleration.





Using the following formulas, you can solve any 1D motionproblem with constant acceleration; just learn them byheart:

a = const.

v = v0 + atx = x0 + v0t + 1

2at2

v2 = v20 + 2a (x − x0)





1D Motion with Zero Acceleration

Zero-acceleration means constant-velocity. The followingare straightforward:

a = 0

v = v0

x = x0 + v0t





Free-Fall Acceleration

All the objects near the surface of the earth gravitatesalways towards the earth because of the downwardgravitational force. This is free-fall acceleration (or,gravitational acceleration).

In all free-fall problems, we neglect the effects of air.

The value of free-fall acceleration near the surface of theearth is almost constant and given as

g = 9.80 m/s2





To solve free-fall problems we label the vertical axis they -axis, with its upward positive direction. Also, we take thedownward acceleration to be a = −g in our formulas:

a = −g

v = v0 − gty = y0 + v0t − 1

2gt2

v2 = v20 − 2g (y − y0)





If an object initially at rest is dropped off an elevation of hand hits the ground in time t , it is easy to show that

h = 12gt2


chapter 2 | motion along a straight line

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