On a $v-t$ graph, which method correctly determines the average velocity over a given time interval?

• A. Find the area under the curve and divide it by the total time interval $—ie,\frac{\int v dt}{\Delta t}$.
• B. Subtract the minimum velocity from the maximum velocity $v\_\max -v\_\min$.
• C. Calculate the slope of the line connecting the initial and final points on the graph $\frac{v\_f-v\_i}{t\_f-t\_i}$.
• D. Measure the maximum value of velocity during the interval $v\_\max$.

Answer: (A)

Verified step by step guidance

1. Understand that the average velocity over a time interval is defined as the total displacement divided by the total time taken. On a velocity-time (v-t) graph, displacement corresponds to the area under the velocity curve between the initial and final times.
2. Identify the time interval \( \Delta t = t_f - t_i \) over which you want to find the average velocity, where \( t_i \) and \( t_f \) are the initial and final times respectively.
3. Calculate the area under the velocity curve between \( t_i \) and \( t_f \). This area represents the displacement \( \Delta x \) during the time interval. Mathematically, this is the definite integral of velocity with respect to time: \[\text{Displacement} = \int_{t_i}^{t_f} v(t) \, dt\]
4. Divide the displacement by the total time interval to find the average velocity: \[\text{Average velocity} = \frac{\int_{t_i}^{t_f} v(t) \, dt}{t_f - t_i}\]
5. Note that other methods like subtracting minimum velocity from maximum velocity or calculating the slope of the line connecting initial and final points on the graph do not give the average velocity over the interval. The correct method involves the area under the curve divided by the time interval.