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.)
the response you submitted has the wrong sign. m/s
(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.)
the response you submitted has the wrong sign. m/s

Explanation:

Step1: Recall average - velocity formula

The average - velocity formula is $v_{avg}=\frac{\Delta x}{\Delta t}$, where $\Delta x=x_f - x_i$ and $\Delta t=t_f - t_i$.

Step2: Determine $x_i$, $x_f$, $t_i$, and $t_f$ from the graph

From the graph, at $t_i = 2.00\ s$, $x_i$ (read from the $x - t$ graph) is approximately $2.0\ m$. At $t_f = 4.00\ s$, $x_f$ is approximately $3.0\ m$.

Step3: Calculate $\Delta x$ and $\Delta t$

$\Delta x=x_f - x_i=3.0 - 2.0=1.0\ m$ and $\Delta t=t_f - t_i=4.0 - 2.0 = 2.0\ s$.

Step4: Calculate average velocity

$v_{avg}=\frac{\Delta x}{\Delta t}=\frac{1.0}{2.0}=0.5\ m/s$.

For part (b):

Step1: Recall instantaneous - velocity concept

The instantaneous velocity at a point on an $x - t$ graph is the slope of the tangent line at that point.

Step2: Determine the slope of the tangent line at $t = 2.00\ s$

By looking at the tangent line at $t = 2.00\ s$ on the graph, if we consider two points on the tangent line (for example, if the tangent line passes through $(2,2)$ and $(3,3)$), the slope $m=\frac{\Delta y}{\Delta x}=\frac{3 - 2}{3 - 2}=1.0\ m/s$.

Answer:

(a) $0.5\ m/s$
(b) $1.0\ m/s$