3.3 Velocity

Definition

Velocity is the rate of change of an object’s displacement with time.

It describes both how fast an object moves and the direction of motion.

Since velocity has both magnitude and direction, it is a vector quantity.

Two objects moving at the same speed can have different velocities if they move in different directions.

SI Unit

The SI unit of velocity is the meter per second (m/s).

A velocity of 5 m/s east, for example, means that an object moves with a speed of 5 m/s in the eastward direction.

Direction of Velocity

The direction of velocity is the same as the direction of motion.

In one-dimensional motion, direction can be represented using positive and negative signs.

If the positive x-direction is chosen to the right:

v = +10 m/s

represents motion to the right, while

v = −10 m/s

represents motion to the left. The sign of velocity indicates the direction of motion, not how fast the object is moving.

Figure 3.3.1-Positive and negative velocities in one-dimensional motion.

In two-dimensional motion, the direction of velocity can be specified using an angle or a direction such as north, south, east or west.

Average Velocity

The velocity of an object does not have to remain constant during a journey. Its speed may change, its direction may change, or both may change.

The average velocity for a time interval is defined as the displacement divided by the time taken.

v→avg=Δx→Δt\vec{v}_{\mathrm{avg}}=\frac{\Delta\vec{x}}{\Delta t}

where:

v→avg=average velocity\vec{v}_{\mathrm{avg}}=\text{average velocity}

Δx→=displacement\Delta\vec{x}=\text{displacement}

Δt=time taken\Delta t=\text{time taken}

Because displacement is a vector quantity, average velocity also has both magnitude and direction.

Average velocity depends on the displacement between the initial and final positions, not on the total distance travelled.

Example 1

A car moves 120 m east in 10 s.

Figure 3.3.2 – A car moving 120 m east in 10 s.

Its average velocity is

v→avg=120m10s\vec{v}_{\mathrm{avg}}=\frac{120\,\mathrm{m}}{10\,\mathrm{s}}

v→avg=12m/s east\vec{v}_{\mathrm{avg}}=12\,\mathrm{m/s\ east}

Example 2

An object moves 3 m east and then 4 m north in a total time of 2.0 s.

Figure 3.3.3 – The displacement of an object moving 3 m east and then 4 m north.

The magnitude of the displacement is 5 m.

The direction of the displacement is

θ=tan−1⁡(43)\theta=\tan^{-1}\left(\frac{4}{3}\right)

θ=53∘ north of east\theta=53^\circ\text{ north of east}

The magnitude of the average velocity is

vavg=5m2.0sv_{\mathrm{avg}}=\frac{5\,\mathrm{m}}{2.0\,\mathrm{s}}

vavg=2.5m/sv_{\mathrm{avg}}=2.5\,\mathrm{m/s}

Average velocity has the same direction as the displacement.

Therefore, the object’s average velocity is 2.5 m/s at 53° north of east.

This example also shows that average velocity is determined by the displacement between the initial and final positions, rather than by the path travelled.

Average Velocity on a Round Trip

Because average velocity depends on displacement, returning to the starting position makes the average velocity zero.

Consider a runner who completes one full 400 m lap in 50 s and returns to the starting position.

Figure 3.3.4 – A runner returning to the starting position after completing a 400 m lap.

The runner travels a total distance of 400 m, but the initial and final positions are the same. Therefore, the displacement is zero.

Δx→=0\Delta\vec{x}=0

The average velocity is

v→avg=050s\vec{v}_{\mathrm{avg}}=\frac{0}{50\,\mathrm{s}}

v→avg=0\vec{v}_{\mathrm{avg}}=0

Although the runner has been moving throughout the lap, the average velocity for the complete journey is zero because there is no displacement between the initial and final positions.

For the same journey, the average speed is 8.0 m/s. This shows that an object can have a non-zero average speed while having zero average velocity.

Instantaneous Velocity

When the velocity of an object changes, its instantaneous velocity is its velocity at a particular moment in time.

Instantaneous velocity describes both how fast the object is moving and its direction of motion at that instant.

The magnitude of instantaneous velocity is equal to the object’s instantaneous speed.

For example, if a car is moving east and its speedometer shows 60 km/h at a particular moment, its instantaneous velocity is 60 km/h east.

If the car changes direction while maintaining the same speed, its instantaneous velocity changes because its direction changes.

Key Points

  • Velocity describes both how fast an object moves and its direction of motion.
  • Velocity is the rate at which displacement changes with time.
  • Velocity is a vector quantity.
  • The SI unit of velocity is the meter per second (m/s).
  • The direction of velocity is the direction of motion.
  • In one-dimensional motion, positive and negative signs can represent opposite directions.
  • Average velocity is displacement divided by the time taken.
  • Average velocity has the same direction as displacement.
  • An object that returns to its starting position has zero average velocity.
  • Average speed depends on distance travelled, while average velocity depends on displacement.
  • Instantaneous velocity is the velocity of an object at a particular moment.
  • The magnitude of instantaneous velocity is instantaneous speed.

← 3.2 Speed  

3.4 Acceleration →