Question

using representations
part c: what quantities should be plotted on a graph if the graph is to have a linear trend and the slope of the best - fit line is to be the launch speed of the dart?
h and t²
part d: plot the quantities stated in part c on the graph below. label the axis with quantities, a scale, and appropriate units. draw a best - fit line.
quantitative analysis
part e: using the best - fit line, determine the speed of the crossbow darts the second they have launched. (hint: carefully calculate the slope and determine the relationship between the quantities you plotted on the graph above and how they relate to the speed of the crossbow darts. use the slope to determine the speed.)

Explanation:

Step1: Recall kinematic equation

For vertical - motion of an object launched vertically upwards, the equation is $H = v_0t-\frac{1}{2}gt^{2}$. If we assume the initial height is 0 and we want a linear relationship where the slope gives the launch - speed $v_0$, we can rewrite it as $H=v_0t + 0t^{2}-\frac{1}{2}gt^{2}$. If we plot $H$ on the y - axis and $t^{2}$ on the x - axis, the equation becomes of the form $y = mx + c$ where $y = H$, $x=t^{2}$, and the slope $m$ is related to the initial velocity.

Step2: Plot the graph

On the x - axis, label it as $t^{2}$ with an appropriate scale (e.g., if the time values range from 0 to 5 s, then $t^{2}$ ranges from 0 to 25 s$^{2}$). Mark evenly spaced intervals. On the y - axis, label it as $H$ (height) with an appropriate scale depending on the range of height values in the experiment. Plot the data points and draw a best - fit line.

Step3: Calculate the slope for speed

The slope of the line $m$ in the graph of $H$ vs $t^{2}$ is related to the initial velocity $v_0$. From the kinematic equation $H = v_0t-\frac{1}{2}gt^{2}$, when we consider the linear relationship between $H$ and $t^{2}$, the slope of the $H$ vs $t^{2}$ graph gives $\frac{v_0}{ \sqrt{2}}$ (derived from the kinematic equations). So, $v_0=\sqrt{2}\times m$. Measure two points $(x_1,y_1)$ and $(x_2,y_2)$ on the best - fit line. The slope $m=\frac{y_2 - y_1}{x_2 - x_1}$. Then calculate the initial speed $v_0$.

Answer:

For part C: Plot height $H$ on the y - axis and $t^{2}$ on the x - axis.
For part D: Follow the steps above to label the axes, choose a scale, plot points, and draw a best - fit line.
For part E: Measure the slope of the best - fit line using two points on the line. Calculate the speed $v_0=\sqrt{2}\times m$ where $m$ is the slope of the $H$ vs $t^{2}$ graph.