2.3 - Average and Instantaneous Acceleration
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00:00 Reviewing Position and Velocity
Velocity is the rate of change in position over time. So whenever an object is changing position, it has some velocity that measures how quickly it is moving. We examined two ways to calculate velocity.
Average Velocity
Instantaneous Velocity
09:26 Calculating Acceleration
Acceleration is the rate of change in velocity over time. Whenever an object’s velocity is changing (maybe it is speeding up or slowing down) it has some acceleration . We can calculate average and instantaneous accelerations:
The units of acceleration are m/s^2.
13:52 Understanding Acceleration
A rocket ship accelerating at 5m/s^2 this means that for every 1 second of time that elapses, its velocity increases by 5m/s. Stated another way, the rocket’s velocity increases by 5 meters per second per every second of time that elapses. That is why the units of acceleration m/s^2 can be read as m/s/s or meters per second per second.
If acceleration is positive then the velocity is becoming more positive. If acceleration is negative then the velocity is becoming more negative. And if acceleration is zero then the velocity is not changing.
15:30 More Complex Acceleration Example
25:21 Velocity – Time Graphs
A line connecting two points on the graph at t=0s and t=5s. The slope of the line 20m/s^2 is equal to the average acceleration between these two points. Likewise we can draw a line connecting the points at t=5s and t=10s. In this case the slope = average acceleration = 40m/s^2
Next we can draw a tangent line to the curve at t=5s. The slope of this line 30m/s^2 is equal to the instantaneous acceleration of the rocket at t=5s. We can verify this is correct by finding the acceleration equation by differentiating the velocity vx and plugging in the value for t=5s.
The average acceleration between two times is equal to the slope of the line connecting the two times on the velocity vs time curve and the instantaneous acceleration at a specific time is equal to the slope of the tangent line of the velocity vs time curve at that time.
34:10 Analyzing Position-Time Graphs
A position vs time graph represents the position of a particle at different times and 6 points on this curve are labeled A through F. On each point we draw a tangent line to the curve. The slope of this tangent line is equivalent to the instantaneous velocity vx at that time.
Each motion diagram shows the position of the particle at that time as well as its velocity. A position-time graph can visually communicate a particle’s position and velocity (via the graph slope) at any time as well as a particle’s acceleration, but it can be more difficult to see the acceleration.
The key to visualizing acceleration on a position-time graph is to understand that acceleration represents a change in velocity over time. And velocity is seen as the slope of the position-time curve. So if the slope is changing, then the velocity is changing and thus there is an acceleration.
In the diagram we sketched out a position-time curve. The tangent to the curve at the marked point and thus the slope of the tangent line is the velocity of the particle at that time. Tangent lines slightly earlier and later in time have a different slope. This change in slope is equivalent to a change in velocity and thus an acceleration. If the slopes are decreasing we have a negative acceleration.
We can now use this understanding to visualize accelerations on the original position-time graph.
At tA we can see the slope of the graph is curving down, meaning the velocity is decreasing and there is a negative acceleration.
At tB the slope of the graph is also curving down and there is a negative acceleration.
In fact the entire first part of the graph up to point C has, what we could call, negative curvature. Thus all points in that section of the graph have a negative acceleration.
At tD and tE the graph is curving upwards. With a positive curvature, the slopes are increasing and we have a positive acceleration.
Points C and F are interesting and represent points where the curvature flips from negative to positive and then back to negative. At these “flipping” points the curvature momentarily becomes zero. Thus there is zero acceleration at tC and tF.
Remember, for a position-time graph:
positive curvature = positive acceleration
negative curvature = negative acceleration
zero curvature = zero acceleration
44:35 Comparing Position-Time and Velocity-Time graphs
Position – Time and Velocity – Time graphs may look similar, but be careful because they are communicating different information. Here is a table comparing the two types of graphs.