Question

a position - time graph for a particle moving along the x - axis is shown in the figure below.
(a) find the average velocity in the time interval t = 2.00 s to t = 4.00 s. (indicate the direction with the sign of your answer.) m/s
did you accidentally divide or take the inverse in your calculation?
(b) determine the instantaneous velocity at t = 2.00 s by measuring the slope of the tangent line shown in the graph. (note that t = 2.00 s is where the tangent line touches the curve. indicate the direction with the sign of your answer.) m/s
enter a number. differs from the correct answer by more than 10%. double check your calculations.
(c) at what value of t is the velocity zero?
your response differs from the correct answer by more than 10%. double check your calculations.

Explanation:

Step1: Recall average - velocity formula

The average velocity $v_{avg}$ over a time interval $\Delta t=t_2 - t_1$ is given by $v_{avg}=\frac{\Delta x}{\Delta t}=\frac{x_2 - x_1}{t_2 - t_1}$, where $x_1$ and $x_2$ are the positions at times $t_1$ and $t_2$ respectively.
For the time interval $t_1 = 2.00\ s$ and $t_2=4.00\ s$, we need to read the positions $x_1$ and $x_2$ from the graph.
From the graph, at $t = 2.00\ s$, $x_1\approx1.5\ m$ and at $t = 4.00\ s$, $x_2\approx3.5\ m$.

Step2: Calculate average velocity

$v_{avg}=\frac{x_2 - x_1}{t_2 - t_1}=\frac{3.5 - 1.5}{4.00 - 2.00}=\frac{2.0}{2.00}=1.0\ m/s$

Step3: Recall instantaneous - velocity method

The instantaneous velocity at a point on a position - time graph is the slope of the tangent line at that point.
For $t = 2.00\ s$, we measure the slope of the tangent line. The tangent line at $t = 2.00\ s$:
We can choose two points on the tangent line. Let's say the tangent line passes through $(t_1,x_1)=(1.00\ s,0.5\ m)$ and $(t_2,x_2)=(3.00\ s,2.5\ m)$
The slope of the tangent line (instantaneous velocity $v$) is $v=\frac{\Delta x}{\Delta t}=\frac{x_2 - x_1}{t_2 - t_1}=\frac{2.5 - 0.5}{3.00 - 1.00}=\frac{2.0}{2.00}=1.0\ m/s$

Step4: Find when velocity is zero

The velocity is zero when the slope of the position - time graph is zero. This occurs when the graph has a horizontal tangent.
By observing the graph, we can see that the slope of the graph is zero at $t\approx6.0\ s$

Answer:

(a) $1.0\ m/s$
(b) $1.0\ m/s$
(c) $6.0\ s$