Question

a block travels along the x - axis and the figure below shows a record of its velocity as a function of time. every gridline along the vertical axis corresponds to 1.50 m/s and each gridline along the horizontal axis corresponds to 4.00 s. (a) determine the average acceleration of the block in the time interval t = 0 to t = 20.0 s. according to the velocity - versus time plot, what is the change in velocity for the time interval of interest? does the velocity change during the time interval? (m/s) determine the average acceleration of the block in the time interval t = 20.0 s to t = 60.0 s. how is the average acceleration related to the change in velocity and change in time? how is the average acceleration over some time interval related to the slope of the line connecting the end - points of the time interval? (b) how is the average acceleration related to the change in velocity and change in time? how is the average acceleration over some time interval related to the slope of the line connecting the end - points of the time interval? determine the average acceleration of the region of the plot where the velocity is increasing at a constant rate (m/s²) that we are interested in the average acceleration over the entire time shown on the plot? (c) determine the average acceleration of the block in the time interval t = 0 to t = 80.0 s. how is the average acceleration related to the change in velocity and change in time? how is the average acceleration over some time interval related to the slope of the line connecting the end - points of the time interval?

Explanation:

Step1: Recall average - acceleration formula

The formula for average acceleration is $a_{avg}=\frac{\Delta v}{\Delta t}$, where $\Delta v = v_f - v_i$ (final velocity - initial velocity) and $\Delta t=t_f - t_i$ (final time - initial time).

Step2: Determine velocities and times for part (a)

From the graph, for the time - interval $t = 0$ s to $t = 20.0$ s. Let's assume the initial velocity $v_i$ at $t = 0$ s and final velocity $v_f$ at $t = 20.0$ s. If we read the velocities from the graph (assuming each vertical grid - line is $1.50$ m/s and each horizontal grid - line is $4.00$ s). Suppose $v_i$ (at $t = 0$) is some value and $v_f$ (at $t = 20$ s) is another value. Calculate $\Delta v=v_f - v_i$ and $\Delta t = 20.0-0=20.0$ s. Then $a_{avg}=\frac{v_f - v_i}{20.0}$.

Step3: Determine velocities and times for part (b)

For the time - interval $t = 20.0$ s to $t = 60.0$ s. Find the initial velocity $v_i$ at $t = 20.0$ s and final velocity $v_f$ at $t = 60.0$ s. Calculate $\Delta v=v_f - v_i$ and $\Delta t=60.0 - 20.0 = 40.0$ s. Then $a_{avg}=\frac{v_f - v_i}{40.0}$.

Step4: Determine velocities and times for part (c)

For the time - interval $t = 0$ s to $t = 80.0$ s. Find the initial velocity $v_i$ at $t = 0$ s and final velocity $v_f$ at $t = 80.0$ s. Calculate $\Delta v=v_f - v_i$ and $\Delta t=80.0 - 0=80.0$ s. Then $a_{avg}=\frac{v_f - v_i}{80.0}$.

Answer:

The average acceleration values need to be calculated by reading the initial and final velocities from the graph for each time - interval and using the formula $a_{avg}=\frac{\Delta v}{\Delta t}$. Without specific values read from the graph, we cannot give numerical answers. But the general method is as described above.