1D Motion, Constant Acceleration

"Aristotle maintained that women have fewer teeth than men; although he was twice married, it never occured to him to verify this statement by examining his wives' mouths"
Bertrand Russell

• In order to discuss motion in general we must first define velocity and acceleration and explain the difference between the average and instantaneous values of these quantities.

• Average Velocity

For a particle traveling between the two points on the graph at right the magnitude of the average velocity is defined by,
, where is the magnitude of the displacement of the particle.
Note that any path between the two points on the graph will result in the same average velocity.  In other words the average velocity tells us nothing of the details of the motion between the initial and final points.
If the motion in question begins and ends at the same location, since is zero, the average velocity is zero.
Average speed is defined as the total distance traveled divided by the time taken.
We must be careful to differentiate between distance and displacement.  Displacement is a vector quantity whose direction is from the initial to final location, whereas distance is a scalar with the usual definition. (See discussion of vectors and scalars to follow)

• Instantanous Velocity

If the initial and final locations are brought closer and closer together, and become smaller and smaller.  The limiting condition,
  defines the instantaneous velocity at a particular location.

The instantanous velocity can also be described as the slope (or tangent) of the curve at a specific location and will in general be different at every point in the path.

Instantanous speed is the magnitude of the instantaneous velocity.

• Units of velocity - m/s (SI) or ft/s (British)

Constant velocity means that instantaneous and average velocities are equal.

• Average and Instantanous Acceleration can be defined in a similar way to average and instantanous velocity.

• Acceleration can be positive or negative, negative acceleration is often called deceleration
• Units of acceleration - m/s² (SI) or ft/s² (British)
• If average and instantaneous accelerations are zero, then the motion is contant velocity.
• Constant acceleration means instantaneous and average acceleration are equal.  In this situation we can obtain a set of equations known as the...

• Kinematic equations for one dimensional constant acceleration motion

• v_f = v_i  +  at
• x  =  v_i t  +  at² /2
• x  =  (v_f ²  -  v_i ² )/2a
• x  =  (v_i  +  v_f )t/2

• Vertical motion under the influence of gravity (ignoring air resistance).

Galileo   determined that

"All freely falling objects have the same acceleration at the same place near the earth's surface"

The value of this acceleration is 9.8 m/s²  (32 ft/s ²) directed towards the centre of the earth.
Since the acceleration is constant, the kinematic equations above may be applied, where a=-g, the v's represent velocities in the y (vertical) direction and x becomes the distance in y (height).


It is said that in order to come to the above conclusion Galileo performed a number of experiments by dropping pairs of different objects off the Leaning Tower of Pisa.  One such pair were reputed to be a feather and a stone. Of course the stone hits the ground first.  However, in the vacuum of the Moon this is not the case, the stone and feather will hit the ground at the same time.  Apollo 15 astronauts David Scott and Jim Irwin actually conducted this experiment on the Moon, the results of which are documented on NASA's Lunar Feather Drop Home Page  and here.  More recently Brian Cox demonstrates Galileo's statement in a vacuum chamber as part of the BBC's Human Universe TV series.

“England and America are two countries divided by a common language”

George Bernard Shaw

Dr. C. L. Davis
Physics Department
University of Louisville
email: c.l.davis@louisville.edu