Question

2.4 section review worksheet (pgs. 46 - 51)
instructions: answer the following questions in complete sentences. show your work. include graphs where necessary.

1. explain how to calculate the slope of a graph.
2. what does the slope of a position versus time graph represent?
3. draw the position versus time graph and the speed versus time graph for an object moving at a constant speed of 2 m/sec.

+ label the axes correctly.
+ show at least 10 seconds of motion.
+ include units on both axes.

1. how can you use a speed versus time graph to find an objects acceleration?

Explanation:

Question 1

Step1: Recall slope formula

The slope $m$ of a graph between two points $(x_1,y_1)$ and $(x_2,y_2)$ is calculated using the formula $m=\frac{y_2 - y_1}{x_2 - x_1}$. You first identify two distinct points on the line - like on a position - time graph, they could be two points representing position and time values. Then you subtract the $y$ - coordinates and divide by the difference in the $x$ - coordinates.

Question 2

The slope of a position versus time graph represents the velocity of the object. Velocity is defined as the rate of change of position with respect to time. Mathematically, $v=\frac{\Delta x}{\Delta t}$, which is the same as the slope formula for the position - time graph where $x$ is position and $t$ is time.

Question 3

For the position - versus - time graph of an object moving at a constant speed of $2$ m/s:

• The position $x$ as a function of time $t$ is given by $x = vt$ (since $v = 2$ m/s, $x=2t$). The $x$ - axis is labeled "Position (m)" and the $t$ - axis is labeled "Time (s)". You can plot points such as $(0,0)$, $(1,2)$, $(2,4)$, $(3,6)$, $(4,8)$, $(5,10)$, $(6,12)$, $(7,14)$, $(8,16)$, $(9,18)$, $(10,20)$ and draw a straight line through them.
• For the speed - versus - time graph of an object moving at a constant speed of $2$ m/s, the speed $v$ is constant over time. The $v$ - axis is labeled "Speed (m/s)" and the $t$ - axis is labeled "Time (s)". The graph is a horizontal line at $v = 2$ m/s for $t$ values from $0$ to at least $10$ s.

Question 4

The acceleration $a$ of an object is the rate of change of speed with respect to time. On a speed - versus - time graph, you can find the acceleration by calculating the slope of the graph. If you have two points $(t_1,v_1)$ and $(t_2,v_2)$ on the speed - versus - time graph, then $a=\frac{v_2 - v_1}{t_2 - t_1}$. Just like finding the slope of any other graph, you take the difference in the $y$ - values (speed values here) and divide by the difference in the $x$ - values (time values here).

Answer:

1. The slope of a graph between two points $(x_1,y_1)$ and $(x_2,y_2)$ is calculated as $m = \frac{y_2 - y_1}{x_2 - x_1}$. Identify two points on the graph and use this formula.
2. The slope of a position - versus - time graph represents the velocity of the object.
3. For the position - versus - time graph, use $x = 2t$ and plot points like $(0,0),(1,2),\cdots,(10,20)$ with $x$ - axis labeled "Position (m)" and $t$ - axis labeled "Time (s)". For the speed - versus - time graph, it is a horizontal line at $v = 2$ m/s with $v$ - axis labeled "Speed (m/s)" and $t$ - axis labeled "Time (s)".
4. The acceleration on a speed - versus - time graph is found by calculating the slope of the graph using the formula $a=\frac{v_2 - v_1}{t_2 - t_1}$ for two points $(t_1,v_1)$ and $(t_2,v_2)$ on the graph.