1. Physics 13 General Physics 1
Kinematics of Motion
Motion in Two or Three Dimensions
MARLON FLORES SACEDON

2. What is Instantaneous velocity? x𝑡-Graph
Motion in Two or Three Dimensions: Instantaneous Velocity and Acceleration
What is Instantaneous velocity?
x𝑡-Graph
Time (𝑡)
Position (𝑥)
𝑥 𝑓
𝑡 𝑓 ∆𝑥
= 𝑥 𝑓 − 𝑥 𝑖
𝑥 𝑖
𝑡 𝑖 ∆𝑡
= 𝑡 𝑓 − 𝑡 𝑖
∆𝑥 ∆𝑡
= 𝑥 𝑓 − 𝑥 𝑖 𝑡 𝑓 − 𝑡 𝑖
𝒗 𝒂𝒗 =
Average velocity

3. What is Instantaneous velocity? x𝑡-Graph
Motion in Two or Three Dimensions: Instantaneous Velocity and Acceleration
What is Instantaneous velocity?
x𝑡-Graph
Time (𝑡)
Position (𝑥)
𝑥 𝑓
𝑥 𝑓
Slope of the secant line ∆𝑥
𝑡 𝑓 ∆𝑥
𝑡 𝑓
Slope of the secant line
Slope of the secant line
𝑥 𝑓 ∆𝑥
𝑡 𝑓
𝑥 𝑖
𝑡 𝑖
Slope of the tangent line ∆𝑥 ∆𝑡 ∆𝑡
∆𝑥 ∆𝑡
= 𝑥 𝑓 − 𝑥 𝑖 𝑡 𝑓 − 𝑡 𝑖
𝒗 𝒂𝒗 =
Average velocity
= lim ∆𝒕→𝟎 ∆𝑥 ∆𝑡
= 𝒅𝒙 𝒅𝒕
𝒗 𝒊𝒏𝒔 = lim ∆𝒕→𝟎 𝒗 𝒂𝒗
Instantaneous velocity
= lim ∆𝒕→𝟎 ∆v ∆𝑡
= 𝒅𝒗 𝒅𝒕
𝒂 𝒊𝒏𝒔 = lim ∆𝒕→𝟎 𝒂 𝒂𝒗
Instantaneous acceleration

4. What is Instantaneous velocity? x𝑡-Graph v𝑡-Graph
Motion in Two or Three Dimensions: Instantaneous Velocity and Acceleration
What is Instantaneous velocity?
x𝑡-Graph
Time (𝑡)
Velocity (𝑣)
v𝑡-Graph
Time (𝑡)
Position (𝑥)
Average acceleration
𝑎 𝑖𝑛𝑠
𝑣 𝑖𝑛𝑠
Average velocity
𝑣 𝑖𝑛𝑠
= 𝑑𝑥 𝑑𝑡
𝑣 𝑖𝑛𝑠
= 𝑑𝑣 𝑑𝑡
𝑎 𝑖𝑛𝑠
∆𝑥 ∆𝑡
= 𝑥 𝑓 − 𝑥 𝑖 𝑡 𝑓 − 𝑡 𝑖
𝒗 𝒂𝒗 =
Average velocity
= lim ∆𝒕→𝟎 ∆𝑥 ∆𝑡
= 𝒅𝒙 𝒅𝒕
𝒗 𝒊𝒏𝒔 = lim ∆𝒕→𝟎 𝒗 𝒂𝒗
Instantaneous velocity
= lim ∆𝒕→𝟎 ∆v ∆𝑡
= 𝒅𝒗 𝒅𝒕
𝒂 𝒊𝒏𝒔 = lim ∆𝒕→𝟎 𝒂 𝒂𝒗
Instantaneous acceleration

5. Position & Velocity Vectors
Motion in Two or Three Dimensions: The Position, Velocity, and Acceleration
Position & Velocity Vectors 𝑧 𝑥 𝑦
𝑟 =𝑥 𝑖 +𝑦 𝑗 +𝑧 𝑘
𝑧 𝑘
𝑦 𝑗
Where:
𝑟 is a position vector
x, y, & z are magnitudes of the components of position vector r
𝑖 , 𝑗 , & 𝑘 are unit vectors 𝑝 𝑟
𝑥 𝑖

6. Motion in Two or Three Dimensions: The Position, Velocity, and Acceleration𝑧 𝑥 𝑦
∆ 𝑟 = 𝑥 2 − 𝑥 1 𝑖 + 𝑦 2 − 𝑦 1 𝑗 + 𝑧 2 − 𝑧 1 𝑘
Where: ∆ 𝑟 is the change in position vector
𝑟 2
𝑝 2
𝑣 𝑎𝑣𝑒 = ∆ 𝑟 ∆𝑡
= 𝑟 2 − 𝑟 1 𝑡 2 − 𝑡 1
∆ 𝑟
= lim ∆𝑡→0 𝑟 2 − 𝑟 1 𝑡 2 − 𝑡 1
𝑣 = lim ∆𝑡→0 𝑣 𝑎𝑣𝑒
𝑟 1
𝑝 1
𝑣 = lim ∆𝑡→0 ∆ 𝑟 ∆𝑡
= 𝑑 𝑟 𝑑𝑡
But: 𝑣 = 𝑣 𝑥 𝑖 + 𝑣 𝑦 𝑗 + 𝑣 𝑧 𝑘
𝑣 𝑥 = 𝑑𝑥 𝑑𝑡
𝑣 𝑦 = 𝑑𝑦 𝑑𝑡
𝑣 𝑧 = 𝑑𝑧 𝑑𝑡
𝑟 1 = 𝑥 1 𝑖 + 𝑦 1 𝑗 + 𝑧 1 𝑘
𝑟 2 = 𝑥 2 𝑖 + 𝑦 2 𝑗 + 𝑧 2 𝑘
𝑣 = 𝑑𝑥 𝑑𝑡 𝑖 + 𝑑𝑦 𝑑𝑡 𝑗 + 𝑑𝑧 𝑑𝑡 𝑘
∆ 𝑟 = 𝑟 2 − 𝑟 1
∆ 𝑟 = 𝑥 2 𝑖 + 𝑦 2 𝑗 + 𝑧 2 𝑘 − 𝑥 1 𝑖 + 𝑦 1 𝑗 + 𝑧 1 𝑘
𝑣= 𝑣 𝑥 𝑣 𝑦 𝑣 𝑧 2

7. Motion in Two or Three Dimensions: The Position, Velocity, and Acceleration
𝑎 = 𝑑 𝑣 𝑑𝑡
= 𝑑 𝑣 𝑥 𝑑𝑡 𝑖 + 𝑑 𝑣 𝑦 𝑑𝑡 𝑗 + 𝑑 𝑣 𝑧 𝑑𝑡 𝑘
𝑎 = 𝑑 2 𝑟 𝑑𝑡 2
= 𝑑 2 𝑥 𝑑𝑡 2 𝑖 + 𝑑 2 𝑦 𝑑𝑡 2 𝑗 + 𝑑 2 𝑧 𝑑𝑡 2 𝑘

8. Motion in Two or Three Dimensions: The Position, Velocity, and Acceleration
Problem # 1

9. Motion in Two or Three Dimensions: The Position, Velocity, and Acceleration
PROBLEM: A robotic vehicle, or rover, is exploring the surface of Mars. The stationary Mars lander is the origin of coordinates, and the surrounding Martian surface lies in the xy-plane. The rover, which we represent as a point, has x- and y-coordinates that vary with time:
𝑥=2.0𝑚− 𝑚 𝑠 𝑡 2
𝑦= 1.0 𝑚 𝑠 𝑡 𝑚 𝑠 𝑡 3
(a) Find the rover’s coordinates and distance from the lander at t = 2.0 s. (b) Find the rover’s displacement and average velocity vectors for the interval t = 0.0 s to t = 2.0 s. (c) Find a general
expression for the rover’s instantaneous velocity vector 𝑣 . Express 𝑣 at t = 2.0 s in component form and in terms of magnitude and direction.
(a) solution
@ 𝑡=2.0𝑠
(b) solution
∆ 𝑟 = 𝑟 2 − 𝑟 1
@ 𝑡=0𝑠
= 1.0𝑚 𝑖 + 2.2𝑚 𝑗 − 2.0𝑚 𝑖
𝑥=2.0𝑚− 𝑚 𝑠 𝑡 2
𝑥=2.0𝑚− 𝑚 𝑠 𝑡 2
=− 1.0𝑚 𝑖 + 2.2𝑚 𝑗
=2.0𝑚− 𝑚 𝑠 𝑠 2
=2.0𝑚− 𝑚 𝑠 𝑠 2
𝑣 𝑎𝑣𝑒 = ∆ 𝑟 ∆𝑡
= − 1.0𝑚 𝑖 + 2.2𝑚 𝑗 2.0𝑠−0𝑠
=1.0𝑚
=2.0𝑚
𝑦= 1.0 𝑚 𝑠 𝑡 𝑚 𝑠 𝑡 3
𝑦= 1.0 𝑚 𝑠 𝑡 𝑚 𝑠 𝑡 3
=− 0.5𝑚 𝑖 + 1.1𝑚 𝑗
= 1.0 𝑚 𝑠 2𝑠 𝑚 𝑠 𝑠 3
= 1.0 𝑚 𝑠 0𝑠 𝑚 𝑠 𝑠 3
= 1.0 𝑚 𝑠 2𝑠 𝑚 𝑠 𝑠 3 =0
=2.2𝑚
@ 𝑡=0𝑠:
𝑟 1 = 2.0𝑚 𝑖
𝑟= 𝑥 2 + 𝑦 2
= 𝑥 1.0𝑚 𝑚 2
=2.4𝑚
@ 𝑡=2.0𝑠:
𝑟 2 = 1.0𝑚 𝑖 + 2.2𝑚 𝑗

10. Motion in Two or Three Dimensions: The Position, Velocity, and Acceleration
@ 𝑡=2 𝑠
PROBLEM: A robotic vehicle, or rover, is exploring the surface of Mars. The stationary Mars lander is the origin of coordinates, and the surrounding Martian surface lies in the xy-plane. The rover, which we represent as a point, has x- and y-coordinates that vary with time:
𝒓 𝟐
∆ 𝒓
𝑥=2.0𝑚− 𝑚 𝑠 𝑡 2
𝑦= 1.0 𝑚 𝑠 𝑡 𝑚 𝑠 𝑡 3
(a) Find the rover’s coordinates and distance from the lander at t = 2.0 s. (b) Find the rover’s displacement and average velocity vectors for the interval t = 0.0 s to t = 2.0 s. (c) Find a general
expression for the rover’s instantaneous velocity vector 𝑣 . Express 𝑣 at t = 2.0 s in component form and in terms of magnitude and direction.
𝒓 𝟏
@ 𝑡=0
(a) solution
@ 𝑡=2.0𝑠
(b) solution
∆ 𝑟 = 𝑟 2 − 𝑟 1
@ 𝑡=0𝑠
= 1.0𝑚 𝑖 + 2.2𝑚 𝑗 − 2.0𝑚 𝑖
𝑥=2.0𝑚− 𝑚 𝑠 𝑡 2
𝑥=2.0𝑚− 𝑚 𝑠 𝑡 2
=− 1.0𝑚 𝑖 + 2.2𝑚 𝑗
=2.0𝑚− 𝑚 𝑠 𝑠 2
=2.0𝑚− 𝑚 𝑠 𝑠 2
𝑣 𝑎𝑣𝑒 = ∆ 𝑟 ∆𝑡
= − 1.0𝑚 𝑖 + 2.2𝑚 𝑗 2.0𝑠−0𝑠
=1.0𝑚
=2.0𝑚
𝑦= 1.0 𝑚 𝑠 𝑡 𝑚 𝑠 𝑡 3
𝑦= 1.0 𝑚 𝑠 𝑡 𝑚 𝑠 𝑡 3
=− 0.5𝑚 𝑖 + 1.1𝑚 𝑗
= 1.0 𝑚 𝑠 2𝑠 𝑚 𝑠 𝑠 3
= 1.0 𝑚 𝑠 0𝑠 𝑚 𝑠 𝑠 3
= 1.0 𝑚 𝑠 2𝑠 𝑚 𝑠 𝑠 3 =0
=2.2𝑚
@ 𝑡=0𝑠:
𝑟 1 = 2.0𝑚 𝑖
𝑟= 𝑥 2 + 𝑦 2
= 𝑥 1.0𝑚 𝑚 2
=2.4𝑚
@ 𝑡=2.0𝑠:
𝑟 2 = 1.0𝑚 𝑖 + 2.2𝑚 𝑗

11. Motion in Two or Three Dimensions: The Position, Velocity, and Acceleration
𝑥=2.0𝑚− 𝑚 𝑠 𝑡 2
@ 𝑡=2.0𝑠
@ 𝑡=0
@ 𝑡=2 𝑠
𝒓 𝟐
𝒓 𝟏
∆ 𝒓
𝑣=1.64𝑚/𝑠
𝑣 𝑦
52.4 𝑜
−𝑣 𝑥
PROBLEM: A robotic vehicle, or rover, is exploring the surface of Mars. The stationary Mars lander is the origin of coordinates, and the surrounding Martian surface lies in the xy-plane. The rover, which we represent as a point, has x- and y-coordinates that vary with time:
𝑥=2.0𝑚− 𝑚 𝑠 𝑡 2
𝑦= 1.0 𝑚 𝑠 𝑡 𝑚 𝑠 𝑡 3
(a) Find the rover’s coordinates and distance from the lander at t = 2.0 s. (b) Find the rover’s displacement and average velocity vectors for the interval t = 0.0 s to t = 2.0 s. (c) Find a general
expression for the rover’s instantaneous velocity vector 𝑣 . Express 𝑣 at t = 2.0 s in component form and in terms of magnitude and direction.
𝑣 = −0.50 𝑚 𝑠 2 𝑡 𝑖 + 1 𝑚 𝑠 𝑚 𝑠 𝑡 2 𝑗
(c) solution
𝑣 = 𝑣 𝑥 𝑖 + 𝑣 𝑦 𝑗 + 𝑣 𝑧 𝑘
𝑣 𝑥 = −0.50 𝑚 𝑠 2 𝑡
𝑣 = 𝑑𝑥 𝑑𝑡 𝑖 + 𝑑𝑦 𝑑𝑡 𝑗 + 𝑑𝑧 𝑑𝑡 𝑘
𝑣 𝑦 =1 𝑚 𝑠 𝑚 𝑠 𝑡 2
= 𝑑 2.0𝑚− 𝑚 𝑠 𝑡 2 𝑑𝑡
𝑣 𝑥 = 𝑑𝑥 𝑑𝑡
= −0.50 𝑚 𝑠 2 𝑡
@ t=2s: 𝑣 𝑥 = −0.50 𝑚 𝑠 =−1.0𝑚/𝑠
𝑣 𝑦 =1 𝑚 𝑠 𝑚 𝑠 =1.3𝑚/𝑠
= 𝑑 𝑚 𝑠 𝑡 𝑚 𝑠 𝑡 3 𝑑𝑡
𝑣 𝑦 = 𝑑𝑦 𝑑𝑡
=1 𝑚 𝑠 𝑚 𝑠 𝑡 2
𝑣= 𝑣 𝑥 𝑣 𝑥 2
= −1.0𝑚/𝑠 𝑚/𝑠 2
=1.64𝑚/𝑠
= 𝑇𝑎𝑛 −
𝛾= 𝑇𝑎𝑛 −1 𝑣 𝑦 𝑣 𝑥
= 52.4 𝑜
𝑣 𝑧 = 𝑑𝑧 𝑑𝑡
= 𝑑 0 𝑑𝑡 =0

12. Motion in Two or Three Dimensions: The Position, Velocity, and Acceleration

13. Motion in Two or Three Dimensions: The Position, Velocity, and Acceleration

14. Motion in Two or Three Dimensions: The Position, Velocity, and Acceleration

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