Estimating Speed

According to a rule-of-thumb, every five seconds betweena lightning flash and the following thunder gives the distanceof the storm in miles. Assuming that the flash of light arrivesin essentially no time at all, estimate the speed of sound in m/sfrom this rule.

1 mi 1609 m 322 m/s

5 s 1 mi

Kinematics in One Dimension

MECHANICS comes in two parts:

kinematics: motion (displacement, time, velocity)x, t, v, a

dynamics: motion and forcesx, t, v, a, p, F

Kinematics in One Dimension

person trainvelocity wrt ground vel vel

5 km/hr 80 km/hr

85 km/hr

Displacement - difference in location; length

consider a coordinate system (for 1-D, it isa number line or single axis).

any difference in locations is a displacement

2 1

30 m 10 m

20 m

x x x

Velocities

0

average speed:

instant speed: limt

xv

t

x dxv

t dt

Average velocity - over the trip, or distance, or time

Instantaneous velocity - right now speed

An airplane travels 2100 km at a speed of 800 km/h, and thenencounters a tailwind that boosts its speed to 1000 km/h for thenext 1800 km.

What was the total time for the trip?

What was the average speed of the plane for this trip?

d = 2100 km + 1800 km = 3900 km

2100 km 1800 kmt = + = 4.43 hr

800 km/hr 1000 km/hr

v = d/t = 881 km/hr

Acceleration

How to express a change in velocity?

Again, two kinds of acceleration:

0

average acceleration:

instant acceleration: limt

va

t

v dva

t dt

Kinematics defined by - x, t, v, a

x displacementt timev velocity

0lim

t

xv

tx dx

vt dt

a acceleration

2

20lim

t

va

t

v dv d xa

t dt dt

An automobile is moving along a straight highway, andthe driver puts on the brakes. If the initial velocity isv1 = 15.0 m/s and it takes 5.0 s to slow to v2 = 5.0 m/s,what is the car’s average acceleration?

From the definition for average acceleration:

2 1

2

5.0 m/s 15.0 m/s

5.0 s

2.0 m/s

v va

t

Motion at Constant Acceleration

kinematics - x, t, v, a

How are these related?

For simplicity, assume that the acceleration is constant:

a = const

0

0

0

va

tv v

t

v v a t

v v a t

Consider someacceleration:

The resultingvelocity:

0

00

0 0 00 0

210 0 2

2

2 2

x xxv

t t

v vx x v t v

v v v v atx x t x t

x x v t a t

For a constantacceleration:

Realize adisplacement:

0 00

0 00

2 20

0

2 20 0

2

2

2

2

v v v vx x vt v t

av v v v

xa

v vx

a

v v a x x

How about an equation of motion without time?

0

00

210 0 2

2 20 0

02

2

v v a t a const

v vv t

x x v t a t

v v a x x

Try It!

Consider an airport runway. A light aircraft must reacha speed of 100 km/hr (27.8 m/s) to lift off. It canaccelerate at 2.00 m/s2.

A) If the runway is 150 m long, can the airplane takeoff?

B) If it cannot take off, how long of a runway wouldbe required?

Try It!

2 20

2 22 20

2

2

27.8 m/s 0 m/s

2 2 2.0 m/s

193 m

v v a d

v vd

a

Doing part B) first:

For part A), runway length is not sufficient.

Read the problemDraw a diagramList what is known and what is wantedWhat physics principles are appropriateList relevant equations and their applicability

(may have to derive the best equation)Calculate the requested quantityMake an estimate - are the results reasonableBalancing units can serve as another check

A car speeding at 80 mi/hr passes a stationary police car.The police car immediately gives pursuit. If the speedingcar remains at a constant velocity, and the police car canmaintain a constant acceleration of 4.5 m/s2, how long isrequired to catch the speeder and how fast is the policecar traveling?

vs = 80 mi/hr = 35.8 m/sap = 4.5 m/s2 = 10.0 mi/hr-s

Seeking to catch the malefactor:

212

212

212

2

2

speeder police

0

m2 35.82 s0 s 15.9 s

m4.5

s

4.5 m/s 16 s 72 m/s 160 mi/hr

S S P P

S P

P S

S

P

P P

x v t x a t

v t a t

a t v t

vt t

a

v a t

If the one-dimensional motion is vertically oriented…

Try a = g (9.807 m/s2 or 32.17 ft/s2 , down)

Galileo derived kinematics based on experiments.Concerning the motion of falling objects,

all objects fall with the same constant acceleration

In the absence of air resistance, regardless of the sizeor mass, all objects fall with the same acceleration g.

A ball is dropped from a tower that is 70.0 m in height.How far will it have fallen in 1.00 s, 2.00 s, and 3.00 s?How long will it take to reach the ground?

212

1 1

2 2

3 3

2

1.00 s; y = 4.90 m

2.00 s; y = 19.6 m

3.00 s; y = 44.1 m

2 2 70 m3.8 s

9.8 m/s

y g t

t

t

t

yt

g

A person throws a ball upward with an initial velocityof 15.0 m/s. How long will the ball take to be caught?

210 2

210 2

10 2

02

2 22 20

2

0

2 2 15.0 m/s0 3.06 s

9.8 m/s

0 m/s 15.0 m/s11.5 m

2 2 ( 9.8 m/s )h

y v t a t a g

v t g t

v g t t

vt t

g

v vy

a