Chapter 4 summary
 Position vector: ( 𝑟⃗ =𝑥𝑖̂ + 𝑦𝑗̂ + 𝑧𝑘̂ )
 Displacement : change in position vector ( ∆𝑟⃗ =∆𝑥𝑖̂ + ∆𝑦𝑗̂ + ∆𝑧𝑘̂ )
∆𝑟⃗

Average velocity ( 𝑣⃗ avg =

Instantaneous velocity ( 𝑣

Average velocity ( 𝑎
⃗ avg =

Instantaneous acceleration (a =
∆𝑡
)
𝑑𝑟⃗
= 𝑑𝑡 )
⃗⃗
∆𝑣
∆𝑡
)
⃗⃗
𝑑𝑣
𝑑𝑡
)
Exercises:
1. Find the position vector ( 2m ,-1m,-3m )
2. A particle moves through displacement
[(2𝑚)𝑖̂ + (6𝑚 )𝑗̂ − (4𝑚)𝑘̂] in (2 s ) find the average
velocity ??
3. The x and y component of an object moving in xy plane
are given as( x=2t3 – 2t2 +t) m and (y=3t2+2t) m
determine :at ( t=2 s)
 The position vector
 Velocity vector
 Acceleration vector
4. At t=0 the velocity of a particle is measured to be 3 m/s
along the negative y- axis and its acceleration is
a=( 2𝑖̂ − 4𝑗̂) (m/s2)
Find the velocity vector as a function of time ??
Projectile motion :
A projectile is :
 A particle moving in the x-y plane
 Have an initial velocity
 It have the free fall acceleration (g)
Projectile motion : the motion of a projectile
 Vi x = vi cos 𝜃 ( ‫ =)ثابته ال تتغير طوال فترة الحركة‬vx at any time
 Vi y = vi sinθ
 Range : horizontal distance (R)
 R=
𝑣𝑖 2 𝑠𝑖𝑛 2𝜃
𝑔
 Maximum range at ( 𝜃= 45°)
 R max=
𝑣𝑖 2
𝑔
 Maximum height ( H =
(𝑣𝑖 sin 𝜃)2
2𝑔
‫أو يمكن استخدام قوانين الحركة‬:
Vertical( ‫↓)حركة عمودية لألسفل‬
Vertical( ‫↑)حركة عمودية لألعلى‬
Vf = vi+ gt
Vf = vi- gt
𝟏
∆𝒚 =vi t + 𝟐gt2
Vf2 = vi2+ 2g ∆𝒚
𝟏
∆𝒚 =𝟐( vf+ vi ) t
𝟏
∆𝒚 =vi t - 𝟐gt2
Vf2 = vi2- 2g ∆𝒚
𝟏
∆𝒚 =𝟐( vf+ vi ) t
 Thrown horizontally ( ‫ = 𝜃 )قذف افقيا‬0
 ‫عند اقصى ارتفاع‬vy =0 ‫لكن‬
 vx
‫لها قيمة وهي نفسها السرعة السينية االبتدائية‬
‫ في حركة المقذوفات االلتزام بأمثلة الكتاب‬: ‫مالحظة‬
Uniform circular motion :
 it travels around a circle path
 the tangential velocity of a particle has constant magnitude and
change direction continuously .
 the center petal acceleration has constant magnitude and
towards at the center of the circle
 the velocity and acceleration are perpendicular to each other
 a=
𝑣2
𝑟
a=center petal acceleration(m/s2)
v=velocity (m/s)
r= radius (m) ( ‫) نصف القطر‬
 period of revolution : the time taken to complete one
revolution
2𝜋𝑟
𝜏=
𝑣
 Frequency :the number of revolution per second
1
𝑓=
𝜏
𝑓=
𝑣
2𝜋𝑟