Kinematics in One Dimension

S123: Physics for Engineers
Kinematics in one Dimension
The motion of objects—baseballs, automobiles, joggers, and even the Sun and
Moon—is an obvious part of everyday life. It was not until the sixteenth and seventeenth
centuries that our modern understanding of motion was established. Many individuals
contributed to this understanding, particularly Galileo Galilei (1564–1642) and Isaac
Newton (1642–1727).
The study of the motion of objects, and the related concepts of force and energy,
form the field called mechanics. Mechanics is customarily divided into two parts: kinematics,
which is the description of how objects move, and dynamics, which deals with force and why
objects move as they do.
Displacement
Displacement is how far the object is from its starting point. That is,
∆𝑥 = 𝑥𝑓 − 𝑥0
Where: Δx is displacement, xf is the final position, and x0 is the initial position.
Average Speed
The average speed of an object is defined as the total distance traveled along its path
divided by the time it takes to travel this distance:
𝐭𝐨𝐭𝐚𝐥 𝐝𝐢𝐬𝐭𝐚𝐧𝐜𝐞 𝐭𝐫𝐚𝐯𝐞𝐥𝐞𝐝

𝐚𝐯𝐞𝐫𝐚𝐠𝐞 𝐬𝐩𝐞𝐞𝐝 =
𝐭𝐢𝐦𝐞 𝐞𝐥𝐚𝐩𝐬𝐞𝐝
Speed has no direction; that is, speed is a scalar.
Average Velocity
𝐝𝐢𝐬𝐩𝐥𝐚𝐜𝐞𝐦𝐞𝐧𝐭 𝐛𝐞𝐭𝐰𝐞𝐞𝐧 𝐭𝐰𝐨 𝐩𝐨𝐢𝐧𝐭𝐬
𝐚𝐯𝐞𝐫𝐚𝐠𝐞 𝐯𝐞𝐥𝐨𝐜𝐢𝐭𝐲 = 𝐯̅ =
𝐭𝐢𝐦𝐞 𝐞𝐥𝐚𝐩𝐬𝐞𝐝 𝐛𝐞𝐭𝐰𝐞𝐞𝐧 𝐭𝐰𝐨 𝐩𝐨𝐢𝐧𝐭𝐬
∆𝒙 𝒙𝟐 − 𝒙𝟏
̅=
𝒗 =
∆𝒕 𝒕𝟐 − 𝒕𝟏
It is important to note that the average velocity is a vector and can be negative, depending
on positions x1 and x2.
Instantaneous Velocity
The quantity that tells us how fast an object is moving anywhere along its path is the
instantaneous velocity, usually called simply velocity. The instantaneous velocity of an object
is the limit of the average velocity as the elapsed time approaches zero, or the derivative of x
with respect to t:
∆𝑥
𝑣 = lim
∆𝑡→0 ∆𝑡
𝑑
𝑣 (𝑡 ) = 𝑥 (𝑡 )
𝑑𝑡
Note that the instantaneous speed always equals the magnitude of the
instantaneous velocity. Why? Because distance traveled and the magnitude of the
displacement become the same when they become infinitesimally small.
Tarlac State University | Prepared by: Engr. A. M. Villanueva Page 1

Acceleration
Acceleration specifies how rapidly the velocity of an object is changing. Average

acceleration is defined as the change in velocity divided by the time taken to make this
change:
𝐜𝐡𝐚𝐧𝐠𝐞 𝐨𝐟 𝐯𝐞𝐥𝐨𝐜𝐢𝐭𝐲
𝐚𝐯𝐞𝐫𝐚𝐠𝐞 𝐚𝐜𝐜𝐞𝐥𝐞𝐫𝐚𝐭𝐢𝐨𝐧 =
𝐭𝐢𝐦𝐞 𝐞𝐥𝐚𝐩𝐬𝐞𝐝
𝑣2 − 𝑣1 ∆𝑣
𝑎̅ = =
𝑡2 − 𝑡1 ∆𝑡
The instantaneous acceleration, a, or acceleration at a specific instant in time, is obtained

using the same process as instantaneous velocity.
∆𝑣
𝑎 = lim
∆𝑡→0 ∆𝑡
𝑑
𝑎 (𝑡 ) = 𝑣 (𝑡 )
𝑑𝑡
Here ∆𝑣 is the very small change in velocity during the very short time interval ∆𝑡.
Motion at Constant Acceleration
Use the definitions of average velocity and acceleration to derive a set of valuable
equations that relate x, v, a, and t when a is constant, allowing us to determine any one
of these variables if we know the others.
The average velocity during the time interval t – t0 will be:
∆𝑥 𝑥 − 𝑥0 𝑥 − 𝑥0
𝑣̅ = = =
∆𝑡 𝑡 − 𝑡0 𝑡
since we chose initial time to be zero, as if time is measured with a stopwatch, elapsed
time is Δt = tf − t0 . Taking t0 = 0 means that Δt = tf = t , the final time on the stopwatch.
 Displacement and Position from Velocity

To get our first two equations, we start with the definition of average velocity:
Substituting the simplified notation for Δx and Δt yields
Solving for x gives us

x = x0 +𝑣̅ t ,
where the average velocity is

𝒗𝟎 + 𝒗
̅=
𝒗
𝟐
The equation reflects the fact that when acceleration is constant, 𝑣̅ is

just the simple average of the initial and final velocities.

 Solving for Final Velocity from Acceleration and Time

We can derive another useful equation by manipulating the definition of acceleration:
Substituting the simplified notation for Δv and Δt gives us
, (constant a)
Solving for v yields v = v0 + at (constant a).
 Solving for Final Position with Constant Acceleration

We can combine the previous equations to find a third equation that allows us to
calculate the final position of an object experiencing constant acceleration. We start
with
Adding v0 to each side of this equation and dividing by 2 gives
Since for constant acceleration, we have
Now we substitute this expression for into the equation for displacement,
, yielding
x = x0 + v0t+0.5at2, (constant a).
 Solving for Final Velocity from Distance and Acceleration

A fourth useful equation can be obtained from another algebraic manipulation of
previous equations. If we solve v = v0 + at for t, we get
Substituting this and into , we get
v2 = v02 + 2a(x − x0) (constant a).
Summary of Kinematic Equations (constant a)

Freely Falling Objects
For free fall, Galileo postulated that all objects would fall with the same constant
acceleration in the absence of air or other resistance. He showed that this postulate
predicts that for an object falling from rest, the distance traveled will be proportional to
the square of the time, that is d α t2.
Galileo’s specific contribution to our understanding of the motion of falling objects can
be summarized as follows: at a given location on the Earth and in the absence of air
resistance, all objects fall with the same constant acceleration.
Kinematic Equations for Objects in Free Fall

We assume here that acceleration equals −g (with the positive direction upward).
Finding Velocity and Displacement from Acceleration
Let’s begin with a particle with an acceleration a(t) which is a known function of time. Since
the time derivative of the velocity function is acceleration,
we can take the indefinite integral of both sides, finding
where C1 is a constant of integration. Since
the velocity is given by
Similarly, the time derivative of the position function is the velocity function,
Thus, we can use the same mathematical manipulations we just used and find
where C2 is a second constant of integration.

Kinematics in One Dimension

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S123: Physics for Engineers

Kinematics in one Dimension

Where: Δx is displacement, xf is the final position, and x0 is the initial position.

𝐭𝐨𝐭𝐚𝐥 𝐝𝐢𝐬𝐭𝐚𝐧𝐜𝐞 𝐭𝐫𝐚𝐯𝐞𝐥𝐞𝐝

Speed has no direction; that is, speed is a scalar.

Tarlac State University | Prepared by: Engr. A. M. Villanueva Page 1

Acceleration specifies how rapidly the velocity of an object is changing. Average

The instantaneous acceleration, a, or acceleration at a specific instant in time, is obtained

Motion at Constant Acceleration

 Displacement and Position from Velocity

Substituting the simplified notation for Δx and Δt yields

Solving for x gives us

where the average velocity is

The equation reflects the fact that when acceleration is constant, 𝑣̅ is

Tarlac State University | Prepared by: Engr. A. M. Villanueva Page 2

 Solving for Final Velocity from Acceleration and Time

Substituting the simplified notation for Δv and Δt gives us

Solving for v yields v = v0 + at (constant a).

 Solving for Final Position with Constant Acceleration

Adding v0 to each side of this equation and dividing by 2 gives

Since for constant acceleration, we have

x = x0 + v0t+0.5at2, (constant a).

 Solving for Final Velocity from Distance and Acceleration

Substituting this and into , we get

v2 = v02 + 2a(x − x0) (constant a).

Summary of Kinematic Equations (constant a)

Tarlac State University | Prepared by: Engr. A. M. Villanueva Page 3

Freely Falling Objects

Kinematic Equations for Objects in Free Fall

Finding Velocity and Displacement from Acceleration

we can take the indefinite integral of both sides, finding

where C1 is a constant of integration. Since

the velocity is given by

where C2 is a second constant of integration.

Tarlac State University | Prepared by: Engr. A. M. Villanueva Page 4

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