Question

1. what does the slope on a distance v. time graph tell you?

1. what does the slope on a velocity v. time graph tell you?

1. sketch and label a positive acceleration graph and a negative acceleration graph.

Explanation:

Question 2

In a distance - time graph, the slope is calculated as the change in distance ($\Delta d$) divided by the change in time ($\Delta t$), i.e., $slope=\frac{\Delta d}{\Delta t}$. By the definition of speed (average speed for non - uniform motion and constant speed for uniform motion), speed $v = \frac{\Delta d}{\Delta t}$. So the slope of a distance - time graph represents the speed of the object.

For a velocity - time graph, the slope is given by the change in velocity ($\Delta v$) divided by the change in time ($\Delta t$), or $slope=\frac{\Delta v}{\Delta t}$. From the definition of acceleration, acceleration $a=\frac{\Delta v}{\Delta t}$. Thus, the slope of a velocity - time graph represents the acceleration of the object.

Positive Acceleration Graph (Velocity - Time Graph)

• Axes: The x - axis is labeled "Time (t)" and the y - axis is labeled "Velocity (v)".
• Shape: The graph is a straight line (for constant positive acceleration) or a curve with an increasing slope (for non - constant positive acceleration) that has a positive slope. For example, if we consider constant positive acceleration, we can have a line starting from a non - zero velocity (say $v_0$) at $t = 0$ and going upwards. Let's assume the acceleration $a$ is constant. The equation of velocity as a function of time is $v=v_0 + at$. So at $t = 0$, $v = v_0$, and as $t$ increases, $v$ increases. We can label the graph with the initial velocity $v_0$, the acceleration $a$ (positive), and mark some points like at $t = t_1$, $v=v_0+at_1$.

Negative Acceleration Graph (Velocity - Time Graph)

• Axes: The x - axis is labeled "Time (t)" and the y - axis is labeled "Velocity (v)".
• Shape: The graph is a straight line (for constant negative acceleration) or a curve with a decreasing slope (for non - constant negative acceleration) that has a negative slope. Using the equation $v = v_0+at$, if $a$ is negative (let's say $a=-|a|$), then $v = v_0-|a|t$. At $t = 0$, $v = v_0$, and as $t$ increases, $v$ decreases. We can label the graph with the initial velocity $v_0$, the negative acceleration (or deceleration) $a$ (negative), and mark points like at $t = t_1$, $v = v_0-|a|t_1$.

If we were to sketch the graphs:

• Positive Acceleration (Velocity - Time):
• Draw a set of axes. Label the horizontal axis as "Time (s)" and the vertical axis as "Velocity (m/s)".
• Plot a line that starts at a point (e.g., (0, 2)) and goes up to a point like (5, 12). The slope of this line is $\frac{12 - 2}{5-0}=\frac{10}{5} = 2\ m/s^2$, which is positive acceleration. Label the line with "Positive Acceleration, $a = 2\ m/s^2$".
• Negative Acceleration (Velocity - Time):
• Draw the same set of axes.
• Plot a line that starts at (0, 12) and goes down to (5, 2). The slope of this line is $\frac{2 - 12}{5 - 0}=\frac{- 10}{5}=- 2\ m/s^2$, which is negative acceleration (deceleration). Label the line with "Negative Acceleration, $a=- 2\ m/s^2$".

Answer:

The slope on a distance v. time graph tells you the speed of the object.

Question 3