the Kinematic Equations

…what they are, and how to use them

(1-Dimensional Examples Provided)

01/19

Definition of Kinematics

“Kinematics is the part of Physics where we are describing the motion within a system.”

- Kinematics does not take into consideration why there is motion…simply how something moves—that is to say, in what way something moves, but not worrying about the Forces involved.

02/19

Frame-of-Reference

When you want to analyze the motion of a system, you will need to have a reference-point which you can consider to be unmoving as time goes on.

03/19

Frame-of-Reference

Take, for instance, the speed of a car relative to the surface of the Earth. We know the Earth is always in motion, but for our system, we can consider it to be stationary.

04/19

Position

Position is often denoted by x or d. You’ve used these variables many times before to show distances, lengths, and displacements.

If you know where you started from, then you can use that as your reference point and you create a system whose motion you can now more easily describe.

05/19

PositionCase-in-Point:

– A man runs from some starting point (xi) to some ending point (xf).

How could we find out his displacement if we did not know where he started from?

Simply put, we couldn’t.06/19

PositionThe major equation used in Kinematics for

finding the position of an object as time passes is this:

x=xo+vot+(1/2)at2

In words, this says, “The current position of the object is equal to its initial position added to its initial velocity multiplied by the amount of time it has been in motion added to one half of the acceleration multiplied by the square of the amount of time it has been in motion.”

07/19

Position

That was a mouthful and a half. However, if we apply it to a real-life situation, it will be more easily understood…

08/19

PositionREAL-LIFE SITUATION:

“Da’ud trips as he is trying to climb down the ladder of a fire-truck. What is his position after three (3) seconds?”

We want to know how far Da’ud has fallen after 3 seconds. Our first instinct should be to jot these equations down:

x=? m t=3 s

a= (-) 9.8 m/s209/19

Position

Once we setup our system, we will figure out two other important facts which will make solving Da’ud’s problem a cinch:

As you can see, xo and vo are both

zero-valued. This

simplifies things…

10/19

Position

…looking back at our Position equation from Kinematics, we see:

x=xo+vot+(1/2)at2

BUT, since xo and vo are both ZERO, this

equation becomes simply x=(1/2)at2 !

11/19

Position

Looking back at our system and applying the equation, we see that…

x=(1/2)at2 x= (0.5)*(-9.8 m/s2)*(3 s)2

Solving, we find:

x=-44.1 m

12/19

Position

The negative sign simply means that Da’ud fell (meaning DOWNward).

Unfortunately for him, 44 meters is approximately 14 stories high…

13/19

Velocity

As you know, velocity is “speed with a direction.” Let’s say that we are interested in knowing what Da’ud’s velocity was just at the point of impact with the Earth’s surface (after 3 seconds had passed).

This time, we need another equation…

14/19

Velocity

The Kinematic Equation for velocity is as follows:

v=vo+at

It’s much simpler than the position equation, and reads thus: “The velocity at some point in time is equal to the initial velocity added to the acceleration acting on the object multiplied by the amount of time that the object has been in motion.”

15/19

Velocity

Once again, we can simplify this equation by realizing that the only Force acting on Da’ud during his descent was the Force of Gravity, and this means that the only acceleration he experienced was the acceleration due to gravity (g) and that vo was, again, 0 m/s.

Rehashing the equation, then, gives us:

v=vo+at v=(0 m/s) + (-9.8 m/s2)*(3 s)

Solving… v=29.4 m/s downward16/19

Velocity

To put this value into perspective, let us do a little Dimensional Analysis:

17/19

Velocity

To put this value into perspective, let us do a little Dimensional Analysis:

17/19

Velocity

To put this value into perspective, let us do a little Dimensional Analysis:

17/19

Velocity

To put this value into perspective, let us do a little Dimensional Analysis:

17/19

Velocity

To put this value into perspective, let us do a little Dimensional Analysis:

And so we’re left with (29.4)*(3600)/(1609) mi/hr

= 65.78 mph ….ouch17/19

Looking Ahead

There are other forms of the Kinematic Equations.

The most useful of these is the “Time-Independent” Kinematic Equation:

vf2=vo

2+2a∆x

(This comes in handy when you do not know how long an object accelerates (time!))

18/19

Practice

Try this one on for size: “Escape Speed is the minimum speed an object

needs to completely break free from the gravity of a planet. The escape speed for the Earth is approximately 8,000 m/s. If an object starts from rest and accelerates at 20 m/s2,

how many seconds will it take to reach escape speed?

What distance will it travel during this time?”

19/19