The Kinematic Equations, Exercises of Reasoning

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The Kinematic Equations

Kinematics - (describing how things move)

Scalar (no direction) Vector (w/ direction)

Distance (d) Displacement (  d) Speed (s) Velocity (v) Acceleration (a) How far you travel Change in position (How far you travel in a given direction) How fast you travel How fast you travel (in a given direction) Rate of change of velocity

Kinematics

Describes motion without regard to

what causes it.

Uses equations to represent the

motion of an object in terms of

acceleration (a), initial velocity (v

i

final velocity (v

f

), displacement (Δx)

and time (t).

These five quantities are related by a

group of equations that we call the

BIG FOUR.

KINEMATIC EQUATIONS

Whenever possible, it will be

convenient to place the frame of

reference at the origin x

i

= 0 m when

t

0

= 0 s.

With this assumption, the

displacement

Δx = x

f

– x

i

becomes

Δx = x.

KINEMATICS BIG FOUR

In BIG FOUR, the average velocity

is simply the average of the initial

velocity and the final velocity:

v = ½(v

i

+ v

f

(This is a consequence of the fact

that the acceleration is constant.)

KINEMATICS BIG FOUR

Each of the BIG FOUR equations is missing

one of the five fundamental quantities.

The way you decide which of equation to

use when solving a problem is to

determine which of the fundamental

quantities is missing from the problem – that is, which quantity is neither given nor asked for – and then use the equation that

doesn’t have that variable.

Example The Displacement of a Speedboat

The speedboat in the figure has a

constant acceleration of +2.0 m/s

2

. If

the initial velocity of the boat is

+6.0 m/s, find its displacement after

8.0 seconds.

Example The Displacement of a Speedboat Reasoning Numerical values for the three unknown variables are listed in the data table. We’re asked to determine the displacement x of the speedboat, so it gets the question mark.

We choose Δx = v

i

t + ½at

2

Δx = (6.0 m/s)(8.0 s) + ½(+2.0 m/s

2

)(8.0 s)

2

Δx = 48 m + 64 m = 112 m

**x a v f v i t ? +2.0 m/s 2

• 6.0 m/s 8.0 s**

Example Catapulting a Jet Solving for t, **v f = v i

• at** 62 m/s = 0 m/s + 31 m/s 2 (t) t = 2.0 s Since the time is now known, the displacement can be found by using Δx = v i t + 1/2a(t) 2 : = (0) (2) + ½ (31 m/s 2 ) (2.0 s) 2 = +62 m

If a car’s initial velocity is +25 m/s, and it accelerated at a rate of +7.5 m/s 2 over a period of 8.0 seconds what is the car’s final velocity? v f = v i

• at v f = 25 m/s + (7.5 m/s 2 ) (8.0 s) v f = 85 m/s x a v f v i t +7.5 m/s 2 ? + 25 m/s 8.0 s

A boy sledding down a hill has an initial speed of +12m/s. He continues to speed up and reaches a final velocity of +18m/s after traveling for 12 seconds. What distance does the boy travel. v = Δx/Δt, so Δx = v av (t)

Δx = (v

f

+ v

i

) (t) = (18 + 12) (12 s)

Δx = 180 m

x a v f v i t ? +18 m/s + 12 m/s 12 s

Mr. Billante is notorious for his road rage. After

being cut off he accelerates at a rate of +12.3 m/s

2

for +455 m. As he approaches the man who cut

him off his final velocity is a ludicrous +125 m/s.

What was his initial velocity?

v

f 2

= v

i 2

+ 2aΔx

2

= v

i 2

+ 2 (12.3 m/s

2

) (455 m)

v

i

= +66.6 m/s

**x a v f v i t 455 m +12.3 m/s 2

• 125 m/s? 12 s**