Average Velocity

When an object moves, its position changes with time. To describe how fast this position changes and in which direction, we use the concept of average velocity. Average velocity helps us understand the overall motion of an object during a given time interval.

Average velocity is a vector quantity defined as the total displacement of an object divided by the total time taken.
\text{Average Velocity} = \frac{\text{Displacement}}{\text{Time Interval}}

It depends on both:

Magnitude (how much displacement occurs)
Direction (the direction of displacement)

SI Unit:
The SI unit of average velocity is metre per second (m/s or ms⁻¹).

NOTE: Average velocity can be positive, negative, or zero, depending on the direction of motion.

Formula for Average Velocity

If an object moves from an initial position x₀ at time t₀to a final position x_f at time t_f, then:

\boxed {v_{\text{avg}} = \frac{x_f - x_0}{t_f - t_0}}

When an object moves, its position changes over time. But have you ever wondered how quickly it changes its position or in which direction? To describe this, we use the concept of average velocity. Here, we will explore and understand average velocity.

Graphical Method: Average velocity from the Displacement-Time graph

Average Speed vs Average Velocity

Average Speed	Average Velocity
Scaler Quantity	Vector quantity
Depends on total distance travelled	Depends on total displacement
Direction not considered	Direction is important
Always positive	Can be positive, negative, or zero

Example:If a car travels along a curved path and returns to its starting point:
Average speed is non-zero
Average velocity is zero (because displacement is zero)

When is Average Velocity Zero?

Average velocity becomes zero when "The initial and final positions are the same i.e., total displacement is zero"

Example:
A car moves from point A to point B and returns back to point A.
Even though the car was moving, its average velocity is zero because the net displacement is zero.

Key Points to Remember

Average velocity depends on displacement, not distance
Direction plays an important role
Zero average velocity does not mean the object was at rest
It gives an overall idea of motion over a time interval

Average Speed
Instanteneous Speed
How does instanteneous speed differ from average velocity?

Solved Examples

Question 1. Find the average velocity between t = 1 and t = 4, for the particle which is moving in a plane and whose position is: r = ti + tj.

Given: the initial and final position vectors,
r = ti + tj
The position vector changes with time. The average velocity is given by the formula,
\vec{v} = \frac{\vec{r} - \vec{r'}}{\Delta t}
At t = 1
r = 1i + 1j
At t = 4
r' = 4i + 4j
\Delta t = 3
Plugging the values in this above equation,
\vec{v} = \frac{\vec{r'} - \vec{r}}{\Delta t}\\ = \vec{v} = \frac{4\hat{i} + 4\hat{j} - (\hat{i} + \hat{j})}{3} \\ = \vec{v} = \frac{3\hat{i} + 3\hat{j}}{3}
v_avg= 1i+1j^{^}

Question 2: Find the displacement vector for the particle which is moving in a plane and whose position vectors are : v_i = 3i + 4j and v_f = 5i + 2j

Given: the initial and final position vectors,
v_i = 3i + 4j
v_f = 5i + 2j
The goal is find the displacement vector 'd ' It is given by,
Δd=v_f−v_i
Plugging the values in this above equation,
d=(5i+2j)−(3i+4j^{^})
d=(5−3)i+(2−4)j
d=2i−2j
The displacement vector of the particle is:
d=2i− 2j

Question 3: Find the displacement vector for the particle which is moving in a plane and whose position vectors are, v_i = i + j and v_f= 2i + 5j

Given: the initial and final position vectors,
v_i = i + j
v_f = 2i + 5j
The goal is to find the displacement vector d. It is given by,
Δd=v_f−v_i
d=(2i+5j)−(i+j)
d=(2−1)i+(5−1)j
d=1i+ 4j
The displacement vector of the particle is:
d=i+4j

Question 4: Find the velocity at t = 4, for the particle which is moving in a plane and whose position is given: r = 2t²i + t³j

Given: the initial and final position vectors,
r = 2t²i + t³j
We need to find the velocity at t=4.
The velocity vector is the derivative of the position vector with respect to time,
v=dr/dt
So, let's differentiate the given position vector r=2t²i + t³j with respect to time:
Differentiate with respect to t,
d/dt (2t²i) =4ti
So the velocity vector is:
v=4ti+3t²j
Plugging the values in this above equation,
v=4(4)i+3(4)2j
v=16i+48j

Unsolved Problems

Question 1: Find the velocity at t = 2, for the particle which is moving in a plane and whose position is given :r = ti + 4t²j

Question 2: Find the average velocity between t = 0 and t = 2, for the particle which is moving in a plane and whose position is given : r = 3ti + 3t³j