Speed and Velocity in 1 Dimension

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3.5Speed and Velocity in 1 Dimension


In these revision notes for Speed and Velocity in 1 Dimension, we cover the following key points:

  • The meaning of Speed
  • How to calculate the speed of a moving object?
  • The difference between average and instantaneous speed
  • The meaning of velocity and how does it differs from speed
  • How to calculate the velocity of a moving object?
  • The difference between average and instantaneous velocity
  • Expressing the velocity as the gradient (slope) of the Position vs Time graph

Speed and Velocity in 1 Dimension Revision Notes

When combining the moving distance and time we obtain a Kinematic quantity known as Speed (v).

By definition, Speed (v) is the Distance travelled by an object in a given time (sometimes we say "in the unit of time" instead of "in a given time").

Mathematically, we can write

v = s/t

Since (in the SI system of units) the Displacement is measured in metres and Time in seconds, the unit of speed is

Unit of speed = [metre/second] = [m/s]

The average speed < v > of an object refers to the total distance it travels divided by the time elapsed.

Mathematically, we write

< v > = stotal/ttotal

Speed is a scalar quantity as it is obtained by dividing two scalars: distance and time of motion.

Instantaneous Speed (v) is another important concept used in Kinematics. It shows the actual speed by which an object is moving.

To calculate the instantaneous speed of an object, we take a small portion of distance and divide it by the corresponding (small) time interval. The smaller the distance (and the time interval) used, the more accurately we can measure the speed for that specific time.

The equation of instantaneous speed v is

v = ∆s/∆t

where Δs represents the small distance considered and Δt the time interval during which the event occurs.

When dividing Displacement and Time, we obtain a new physical quantity known as Velocity.

Velocity is a vector quantity and it is denoted by v. Its formula is

Velocity = Displacement/Time

In symbols (in one dimension) the above equation is written as

v = ∆x/t

Since Displacement is measured in metres and Time in seconds, the unit of Velocity is [m/s] just like the unit of Speed.

It is obvious Velocity is a vector quantity as it is obtained by dividing a vector by a scalar.

If we want to calculate the average and instantaneous velocity, the approach is the same as for average and instantaneous speed. Thus, Average Velocity < v > represents the total Displacement divided by the total Time taken. The concept of average velocity is particularly useful when dealing with non-uniform motion.

Mathematically, we have:

< v > = ∆xtot/ttot

As for the Instantaneous velocity v, again we take two neighbouring values for Position and Time. In this way, we obtain one interval for each quantity, namely ∆x and ∆t. Therefore, we obtain the equation

v = ∆x/∆t

A very important rule in Kinematics states that:

The slope (gradient) of the Position vs Time graph gives the Velocity.

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