Defining Average Acceleration ($a$)

In physics, acceleration describes how an object's velocity changes over time. Velocity includes both speed and direction. When an object's velocity is not constant, it is accelerating.

The average acceleration ($a$) is calculated over a specific period. It represents the total change in velocity divided by the total time taken for that change to occur. This gives us a measure of the acceleration averaged over the entire time interval, smoothing out any variations within that interval.

Mathematical Expression for Average Acceleration

To define the average acceleration, we consider:

• The initial velocity ($v_i$): The velocity of the object at the beginning of the time interval.
• The final velocity ($v_f$): The velocity of the object at the end of the time interval.
• The time interval ($t$): The duration over which the velocity changes.

The change in velocity is calculated as the final velocity minus the initial velocity:

$ \Delta v = v_f - v_i $

The average acceleration ($a$) is then found by dividing this change in velocity ($\Delta v$) by the time interval ($t$):

$ a = \frac{\Delta v}{t} $

Substituting the expression for $\Delta v$, we get the formula:

$ a = \frac{v_f - v_i}{t} $

Analyzing the Options for Average Acceleration

Let's examine each provided option based on the definition of average acceleration:

• Option 1: $a = (v_f + v_i) / t$

  This expression incorrectly adds the initial and final velocities instead of finding the difference. The change in velocity is required, not the sum.

• Option 2: $a = (v_f - v_i) \cdot t$

  This option multiplies the change in velocity by the time interval. Average acceleration is defined as the change in velocity *divided* by the time interval, not multiplied.

• Option 3: $a = \Delta x / t^2$

  This expression involves displacement ($\Delta x$) and time squared ($t^2$). While related to motion, this formula does not represent the definition of average acceleration, which specifically relates to the change in velocity over time.

• Option 4: $a = (v_f - v_i) / t$

  This expression correctly represents the change in velocity ($v_f - v_i$) divided by the time interval ($t$). This aligns perfectly with the definition of average acceleration.

• Option 5: (Empty Option)

  This option is not provided.

Conclusion on Average Acceleration Formula

The definition of average acceleration ($a$) is the rate at which velocity changes over a specific time interval ($t$). It is calculated by taking the difference between the final velocity ($v_f$) and the initial velocity ($v_i$), and then dividing the result by the time interval.

Therefore, the correct mathematical expression is:

$ a = \frac{v_f - v_i}{t} $