Kinematics of Motion in a Straight Line One Dimension by Owen Borville November 14, 2025
Kinematics is the study of the motion of an object or objects.
Displacement is the change in position of an object and is measured in length units of meters or feet.
Δx = xf-x0
x0 is the initial position or x1
xf is the final position or x2
Δ means delta or "change in"
A vector is a quantity that has magnitude and direction. A scalar is a quantity that has magnitude only and no direction. Displacement and velocity are vectors, but distance and speed are scalar quantities. Direction is specified by a plus sign or a minus sign for a change to the opposite direction.
Time is measured in seconds (s) (SI units). The time of an event is:
Δt = tf-t0
tf is the final time or t2
t0 is the initial time or t1
Average velocity (av) is the displacement divided by the travel time, measured in meters per second (m/s) (SI units). Velocity is a vector and has direction.
av = Δx/Δt = (xf-x0)/(tf-t0) or (x2-x1)/(t2-t1)
Average speed = (total distance)/(elapsed time)
Instantaneous velocity (v) is the velocity at a specific instant in time or the average velocity for an infinitely small period of time greater than zero but less than a positive real number. Instantaneous speed is the magnitude of the instantaneous velocity. Instantaneous speed is a scalar quantity. Average speed is the total distance traveled divided by the total elapsed time. Velocity has magnitude and direction while speed has only magnitude, so speed can be higher than velocity if the object returns toward the original position. Using calculus, instantaneous velocity v(t) can be found by calculating the derivative of the position function. Instantaneous speed can be calculated by taking the absolute value of instantaneous velocity. The slope of a position-versus-time graph at a specific time gives instantaneous velocity at that time.
Instantaneous velocity = v(t) =dx(t)/dt
Instantaneous speed = |v(t)|
Acceleration is the rate at which velocity changes. SI units for acceleration are meters per second squared. Acceleration is a vector and has magnitude and direction. Acceleration can be caused by either a change in magnitude or the direction of the velocity. Instantaneous acceleration a(t) is the acceleration at a specific instant in time and can be found using calculus by taking the derivative of the velocity function. Instantaneous acceleration is the slope of the velocity-versus-time graph. Deceleration (negative acceleration) is an acceleration with a direction opposite to that of the velocity. Average acceleration (a):
average acceleration= Δv/Δt = (vf-v0)/(tf-to) or (v2-v1)/(t2-t1)
instantaneous acceleration = a(t) = dv(t)/dt
Kinematic equations for motion assuming acceleration is constant and initial time is zero. Initial position and velocity are also zero.
x = x0 + (av)t
av = (v0 + v)/2
v=v0 + at
x = x0 +v0t + 1/2(at)^2
v^2 = v0^2 + 2a(x - x0)
y can be substituted for x to represent vertical motion
Falling objects due to gravity (g):
g = acceleration due to gravity of an object = 9.8 m/s^2
v = v0 - gt
y = y0 + v0t - 1/2gt^2
v^2 = v0^2 - 2g(y - y0)
Calculus offers more tools to solve physics problems than algebra. For example, if acceleration a(t) is known, velocity v(t) and position x(t) can be found using integral calculus (integration). If acceleration is constant, however, the above equations can be used: (v=v0 + at) and (x = x0 +v0t + 1/2(at)^2).
Velocity from acceleration = v(t) = ∫a(t)dt + C1
Position from velocity = x(t) = ∫v(t)dt + C2
Differentiation and integration are inverse operations of each other, much like how taking a base-10 logarithm and exponentiation with base 10 are inverse functions. The relationship between differentiation and integration is formally established by the Fundamental Theorem of Calculus.
Kinematics is the study of the motion of an object or objects.
Displacement is the change in position of an object and is measured in length units of meters or feet.
Δx = xf-x0
x0 is the initial position or x1
xf is the final position or x2
Δ means delta or "change in"
A vector is a quantity that has magnitude and direction. A scalar is a quantity that has magnitude only and no direction. Displacement and velocity are vectors, but distance and speed are scalar quantities. Direction is specified by a plus sign or a minus sign for a change to the opposite direction.
Time is measured in seconds (s) (SI units). The time of an event is:
Δt = tf-t0
tf is the final time or t2
t0 is the initial time or t1
Average velocity (av) is the displacement divided by the travel time, measured in meters per second (m/s) (SI units). Velocity is a vector and has direction.
av = Δx/Δt = (xf-x0)/(tf-t0) or (x2-x1)/(t2-t1)
Average speed = (total distance)/(elapsed time)
Instantaneous velocity (v) is the velocity at a specific instant in time or the average velocity for an infinitely small period of time greater than zero but less than a positive real number. Instantaneous speed is the magnitude of the instantaneous velocity. Instantaneous speed is a scalar quantity. Average speed is the total distance traveled divided by the total elapsed time. Velocity has magnitude and direction while speed has only magnitude, so speed can be higher than velocity if the object returns toward the original position. Using calculus, instantaneous velocity v(t) can be found by calculating the derivative of the position function. Instantaneous speed can be calculated by taking the absolute value of instantaneous velocity. The slope of a position-versus-time graph at a specific time gives instantaneous velocity at that time.
Instantaneous velocity = v(t) =dx(t)/dt
Instantaneous speed = |v(t)|
Acceleration is the rate at which velocity changes. SI units for acceleration are meters per second squared. Acceleration is a vector and has magnitude and direction. Acceleration can be caused by either a change in magnitude or the direction of the velocity. Instantaneous acceleration a(t) is the acceleration at a specific instant in time and can be found using calculus by taking the derivative of the velocity function. Instantaneous acceleration is the slope of the velocity-versus-time graph. Deceleration (negative acceleration) is an acceleration with a direction opposite to that of the velocity. Average acceleration (a):
average acceleration= Δv/Δt = (vf-v0)/(tf-to) or (v2-v1)/(t2-t1)
instantaneous acceleration = a(t) = dv(t)/dt
Kinematic equations for motion assuming acceleration is constant and initial time is zero. Initial position and velocity are also zero.
x = x0 + (av)t
av = (v0 + v)/2
v=v0 + at
x = x0 +v0t + 1/2(at)^2
v^2 = v0^2 + 2a(x - x0)
y can be substituted for x to represent vertical motion
Falling objects due to gravity (g):
g = acceleration due to gravity of an object = 9.8 m/s^2
v = v0 - gt
y = y0 + v0t - 1/2gt^2
v^2 = v0^2 - 2g(y - y0)
Calculus offers more tools to solve physics problems than algebra. For example, if acceleration a(t) is known, velocity v(t) and position x(t) can be found using integral calculus (integration). If acceleration is constant, however, the above equations can be used: (v=v0 + at) and (x = x0 +v0t + 1/2(at)^2).
Velocity from acceleration = v(t) = ∫a(t)dt + C1
Position from velocity = x(t) = ∫v(t)dt + C2
Differentiation and integration are inverse operations of each other, much like how taking a base-10 logarithm and exponentiation with base 10 are inverse functions. The relationship between differentiation and integration is formally established by the Fundamental Theorem of Calculus.