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Description of motion in terms of SPACE and TIME without reference to the masses and forces(„geometry of Physics”)
we ignore agents that caused motion, size, shape, internal structure of a body (a point mass)
On motion
Motion: the most simple observed changean important aim of Physics: procesess in time
Any movement is defined in relation to a frame ofreference ( a stationary coordinate system)
empty space?
Assumptions of classical mechanics:
- size of bodies- velocities
Lec 2 / Physics for Engineers I / M.Mulak IF WUT2
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1ˆ =nA unit vector defines a direction in space
Projection of a vector on a given direction: ˆ ˆ( ) nA A n n= •r r
jikikjkji
kjkijiˆˆˆ,ˆˆˆ,ˆˆˆ0ˆˆ,0ˆˆ,0ˆˆ
=×=×=×
=•=•=•
For a right-handed system:
*Other systems: polar, spherical, cylindrical
kji ˆ,ˆ,ˆ - unit vectors
Cartesian (rectangular) coordinate system
Lec 2 / Physics for Engineers I / M.Mulak IF WUT4
1D kinematics
Position as a function of time: )(txx rr=
average speed∆t
svśr
∆=
r
displacement )()( 12 txtxx rrr−=∆
distance: the total length of the path (always positive scalar) s∆
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Example: average velocity and average speed
10m/s 100m
20m/s przez 15s
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Average velocity
tx
tttxtx
śr ∆∆
=−−
=rrr
r
12
12 )()(v
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instantaneous velocity
dtxd
ttxttx
t
rrrr
=∆−∆+
=→∆
)()(limv0
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Acceleration: average and instantaneous
tttttaśr ∆
∆=
−−
=v)(v)(v
12
12rrr
r
2
2
0
v)(v)(vlimdtxd
dtd
tttta
t
rrrrr
==∆−∆+
=→∆
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1D motion: examples
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Uniformly accelerated motion
0 lub 0 <> aa
2
0 0 0( ) , 2at dxx t x v t v v at
dt= + + = = +
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( ) ( ) ( )x t v t a t⇒ ⇒
Graphical relations
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position vector
),,(ˆˆˆ zyxkzjyixr =++=r
3D motion
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3D kinematics
displacement vector
)()( 12 trtrr rrr−=∆
instantaneous velocity
, , ( , , )x y zdr dx dy dzv v v vdt dt dt dt
= = =
rr
Speed: magnitude (modulus) of velocity
vv r=
(tangent to the path)
)(trr rr= trajectory
Lec 2 / Physics for Engineers I / M.Mulak IF WUT18
Instantaneous acceleration
),,(,, zyxzyx aaadtdv
dtdv
dtdv
dtvda =
==
rr
tangent and normal components
22tnt aaa
dtdva −==
In generalnot tangent to the path
2D & 3D motion
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2
2gth =
Free fall
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2
2gth =
st 2.0=
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From: Bioastronautics Data Book 1973
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Projectile motion
constant g
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„ Anatomy”of a projectile motion
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0
20
0
20 0
0
( cos ) ( )1( sin )
2
2 sin
2 sin sin 2( cos ) =
x v ty x
y v t gt
vtg
v vR vg g
θ
θ
θ
θ θθ
= ⇒
= −
=
=
Max range for 45oθ = 30
v V v ′= +rr r
' | dr R rd t
= +urrr
Principle of relativity of motion
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Key terms
range of validity, model, particle, physical quantity,unit, second, meter, kilogram, uncertainty, error, percent error, significant figures, scientific (powers-of-10) notation, precision, order of magnitude estimates, scalar quantity, vector quantity, magnitude of vector, displacement, parallel vectors, antiparalell vectors, negative of a vector, vector sum (resultant), component vectors, components, unit vector, scalar (dot) product, vector (cross) product, right-hand rule, right-handed system, position vector, average velocity, instantaneous acceleration, projectile motion, trajectory, uniform circular motion, centripetal acceleration, period, non-uniform circular motion, relative velocity, frame of reference.
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1. Draw graphs of the functions x(t), v(t), a(t) for uniform velocity and uniformly accelerated motion.2. Prove that:3. A position vector of a particle depends on time as follows:
find: equation of the path, velocity, speed, acceleration and its magnitude, tangent and normal components of the acceleration of this particle. How would you call such a kind of motion ?4. A position vector of a particle depends on time as follows:find: equation of the path, velocity, speed, acceleration and its magnitude of this particle. Identify and explain the meaning of the constants appearing in the expressions above. What are the dimensions of the constants ? How would you call such a kind of motion ?5. A boat is crossing a river. How should it move relative to the water to:a) cross the river in shortest time (to minimize the time of travel),b) cross the river at shortest path in relation to the ground (to go directly across the river).Is the second (b) way always possible ? 6. A jogger runs his first 100 m at 5 m/s and the second 100 m at 4 m/s. What is her average speed?
=
dtdz
dtdy
dtdx
dtrd ,,
constAwheretAtAr −= ωωω ,)0),sin(),cos((r
Additional problems
