Linear Particle Motion: Position, Velocity, Acceleration
Displacement vs. Distance Traveled
Displacement is the change in position, can be positive or negative
Displacement = Final Position – Initial Position
Distance is always positive.
Total Distance Traveled = Accumulation of all traveled distance
Visual:
A particle goes from
0 G
Distance Traveled
Displacement
0H
0H G
HGH
Velocity vs. Speed
Velocity has direction and magnitude
Speed = velocity , has no direction and is always positive
Example: For a particle moving to the left 5 meters per second, its velocity is ____________ but its speed is _________.
Position, Velocity and Acceleration
Let 𝒔(𝒕)= position, 𝒗(𝒕)= velocity, 𝒂(𝒕)= acceleration
Velocity is the rate of change in position
Acceleration is the rate of change in velocity
Reference Chart:
Let 𝒔(𝒕)= position, 𝒗(𝒕)= velocity, 𝒂(𝒕)= acceleration
Given
Position Function: s(t)
Velocity Function: v(t)
Displacement over interval [a, b]
Final Position – Initial Position
s(b) – s(a)
∫ 𝑣(𝑡) 𝑑𝑡
s(b)
Initial Position + Displacement
𝑏
𝑎
𝑏
Current Position at t = b
𝑠(𝑎) + ∫ 𝑣(𝑡) 𝑑𝑡
𝑎
Total Distance Traveled
Sum of positive distances
traveled between resting points
Average Velocity over time interval [a, b]
𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑠(𝑏) − 𝑠(𝑎)
=
𝑡𝑖𝑚𝑒
𝑡
Instantaneous Velocity at time t = b
𝑠′(𝑏)
At Rest
Particle Moving Right (forward or up)
Particle Moving Left (backward or down)
Velocity is increasing
Velocity is decreasing
Speed
Speed is increasing
Speed is decreasing
∫ 𝑠𝑝𝑒𝑒𝑑 = ∫ |𝑣(𝑡)|𝑑𝑡
The average value =
𝑣(𝑏)
Linear Motion Part 1 (Graph Analysis)
Definite Integral calculates Net Accumulation over an interval.
Example:
Example:
A car travels a straight road obeys the velocity function
per hour and t is measured in hours.
Where velocity is measured in miles
a) Sketch the velocity function
b) Did the car change its direction? At what time?
c) What is the acceleration of the car when it changed direction? What is its velocity?
c) What is the change in the car’s position?
d) What is the total distance traveled?
e) Suppose the car started at 4 miles away in the positive direction, what is its position when t = 5?
Example:
NOTE:
Displacement over 𝑎 < 𝑡 < 𝑏 
Current Position at (t = b) 
Total Distance Traveled 
The above graph shows the velocity v=f(t) of a particle. If the initial position is at y = 2, what is the particle’s position at
time = 1, time = 2, time = 3?
You Try: If the following graph represents velocity.
(a) If the initial position is at y = -3, what is the particle’s position at time = 2, time = 4, time = 6?
(b) What is the total distance traveled?
Linear Motion Part 2 (Function Analysis)
Example: Given the position function
The position function of a particle moving along the x axis is given by s(t) = 3t² - 24t + 36
a) Find the velocity and acceleration functions of the particle.
b) Find the open t-intervals when the particle is moving to the left
c) When is the particle at rest?
d) What is the particle’s position when it’s at rest?
e) What is the displacement of the particle over the interval [0, 8]
f) What is the total distance traveled [0, 8]?
Example: Given the velocity function
Suppose that the velocity 𝑣(𝑡) of an object at time 𝑡 along a straight horizontal line is given by the following function:
𝑣(𝑡) = −𝑡 2 + 4𝑡 − 3
a) For 0 ≤ 𝑡 ≤ 5, when is the object moving to the right?
b) What is the speed of the object at time 𝑡 = 4?
c) If the object is at position 𝑠 = 6 when 𝑡 = 0, what is its position at time 𝑡 = 2?
d) Find the total displacement of the object from 𝑡 = 0 until 𝑡 = 5.
e) Find the total distance traveled by the object from 𝑡 = 0 until 𝑡 = 5.
PRACTICE:
(1) A particle starts at time t = 0 and moves on a number line so that its position at time is given by x(t) = (t - 2)²(t - 6).
a) When is the particle at rest?
b) When does the particle change direction?
c) What is the farthest to the left of the origin the particle moves?
(2) A particle moves along the x –axis so that at time t, its position is given by x(t) = t³ - 6t² + 9t – 11
a) What is the initial velocity of the particle?
b) At what time interval is it moving to the left?
c) What is the total distance traveled from t = 0 to 2?
D) Is the particle’s speed increasing or decreasing at t = 4?
(3) Suppose that the velocity 𝑣(𝑡) of an object at time 𝑡 along a straight horizontal line is given by the following function:
𝑣(𝑡) = −𝑡 2 + 9
a) For 0 ≤ 𝑡 ≤ 6, when did the object change direction?
b) What is the acceleration when the object changes directions?
c) What is the object’s average velocity for 0 ≤ 𝑡 ≤ 6?
d) When is the speed of the object start increasing?
e) Find the total displacement of the object from 𝑡 = 0 until 𝑡 = 6.
f) Find the total distance traveled by the object from 𝑡 = 0 until 𝑡 = 6.
1
3
3
2
(4) A particle is moving along the x-axis in such a way that its position at time t is given by x(t )  t  3t  8t
a) Find all values of t for which the particle is moving to the left.
b) What is its position at t = 3?
c) When t = 3, what is the displacement? What is total distance
d) When t = 3, what is the acceleration of the
traveled?
particle?