Why Average Velocity Uses Two Positions
A particle can move along a line while its position changes over time. To describe its motion across a time interval, compare its position at the beginning of the interval with its position at the end. The resulting rate describes the particle’s overall change in position per unit of elapsed time.
The key word is average. The calculation summarizes the change over the entire interval; it does not say that the particle moved at that rate at every moment. The sign matters too: position is measured relative to a chosen coordinate direction, so a negative average velocity indicates an overall change toward the negative direction.
Here \(s(t_1)\) is the initial position and \(s(t_2)\) is the final position for this interval. The numerator is the signed change in position, and the denominator is the positive amount of elapsed time. Keep the endpoint order consistent: final minus initial in both places.
Average Velocity as a Secant Slope
A position-time graph plots time horizontally and position vertically. At the two times \(t_1\) and \(t_2\), the graph contains the points \((t_1,s(t_1))\) and \((t_2,s(t_2))\). The straight line through these two points is called a secant line. Its slope is the difference in vertical coordinates divided by the difference in horizontal coordinates.
This gives a direct geometric interpretation: average velocity is the slope of the secant line joining the two position-time data points. A positive slope means the later position is greater than the earlier one; a negative slope means it is smaller. A horizontal secant has slope zero, so the initial and final positions are equal, even if the particle may have moved between those times.
Why the Formula Works
The formula follows from the meaning of rate. If a quantity changes by \(\Delta s\) during an elapsed time \(\Delta t\), its average rate of change is \(\Delta s/\Delta t\). For position data, the change is final position minus initial position, while the elapsed time is final time minus initial time.
The points on a position-time graph are \((t_1,s(t_1))\) and \((t_2,s(t_2))\).
The change in position is \(s(t_2)-s(t_1)\), and the change in time is \(t_2-t_1\).
The quotient is both the average rate of change of position and the slope of the secant through the two points.
This reasoning also explains why the units are position units divided by time units. If position is measured in meters and time in seconds, average velocity is measured in meters per second. The units should be included in the final answer, not just the numerical value.
Calculating from Two Position Measurements
When two positions are given directly, first identify which belongs to the earlier time and which belongs to the later time. Subtract the positions in chronological order, then divide by the elapsed time.
Worked Example: Average Velocity from Two Positions
A particle is at position \(5\) meters when \(t=2\) seconds and at position \(-7\) meters when \(t=8\) seconds. Find its average velocity over this interval.
The initial time is \(t_1=2\) seconds, and the final time is \(t_2=8\) seconds. The initial and final positions are \(s(t_1)=5\) meters and \(s(t_2)=-7\) meters.
The average velocity is \(-2\) meters per second. Its negative sign means that the net change in position is toward the negative coordinate direction. The secant slope is also \((-7-5)/(8-2)=-2\), using the points \((2,5)\) and \((8,-7)\).
Check the result by multiplying the average velocity by the elapsed time: \((-2\text{ meters per second})(6\text{ seconds})=-12\text{ meters}\). This matches the change in position, \(-7-5=-12\) meters.
Calculating from a Position Table
A table may list positions at several times. Use only the two rows that give the endpoints of the requested interval. Do not average every listed position: the calculation depends on the position change between the chosen endpoints and the time elapsed between them.
Worked Example: Choosing Endpoints from a Table
A particle’s position is recorded in the table. Find its average velocity from \(t=3\) seconds to \(t=10\) seconds.
| Time (seconds) | Position (meters) |
|---|---|
| 0 | 2 |
| 3 | 11 |
| 7 | 3 |
| 10 | -4 |
At \(t=3\), the position is \(11\) meters. At \(t=10\), the position is \(-4\) meters. Therefore, the change in position is \(-4-11=-15\) meters, and the elapsed time is \(10-3=7\) seconds.
The exact value is \(-15/7\) meters per second. Rounded to the nearest hundredth, it is approximately \(-2.14\) meters per second. The secant connects the points \((3,11)\) and \((10,-4)\), so its slope is \(-15/7\) as well.
Check by multiplying the exact average velocity by the elapsed time: \((-15/7)(7)=-15\) meters. This agrees with the table’s net position change.
Using a Position Formula
If a position formula is given, evaluate it at both endpoints before forming the quotient. This keeps the calculation organized and reduces the chance of substituting an endpoint incorrectly.
Worked Example: Average Velocity from a Position Function
Suppose \(s(t)=t^2-4t+1\) meters, with \(t\) measured in seconds. Find the average velocity from \(t=1\) to \(t=4\) seconds.
Evaluate the position at the initial time:
Evaluate it at the final time:
The position change is \(1-(-2)=3\) meters, and the elapsed time is \(4-1=3\) seconds. Thus,
The corresponding points are \((1,-2)\) and \((4,1)\), whose secant slope is \((1-(-2))/(4-1)=1\). As a check, \(1\text{ meter per second}\) multiplied by \(3\) seconds gives a \(3\)-meter change in position, matching the computed change.
Interpreting the Sign and Units
The sign of average velocity describes the direction of the net change in position over the chosen interval, relative to the coordinate direction. It does not depend simply on whether the positions themselves are positive or negative. For instance, a change from position \(-8\) meters to position \(-3\) meters is positive because \(-3-(-8)=5\) meters.
A zero average velocity means that the final position equals the initial position. It does not, by itself, establish that the particle stayed at rest throughout the interval. The average only uses the endpoint positions.
Units provide another useful check. If position is in feet and time is in seconds, the quotient must be in feet per second. If the calculated units are instead seconds per foot, the changes were divided in the wrong order. When converting units is needed, express both endpoint positions and both times in compatible units before interpreting the result.
Common Mistakes to Avoid
A frequent error is to reverse the order of the times but not the order of the positions. For example, using the data from the first worked example, the incorrect calculation
gives the wrong sign. The numerator uses initial minus final, while the denominator uses final minus initial. Correct the calculation by using the same endpoint order in both differences:
Another mistake is to divide a position by a time rather than divide the change in position by the change in time. In the first example, \(-7/8\) compares the final position with the final time; it does not measure the average velocity from \(t=2\) to \(t=8\). Use both endpoint values in the formula.
Check Your Understanding
For each question, identify the endpoint data before calculating, and include units where they are given.
- A particle is at \(12\) feet when \(t=1\) second and at \(4\) feet when \(t=5\) seconds. What is its average velocity on this interval?
- A position table gives \(s(2)= -3\) meters and \(s(9)=18\) meters. Find the average velocity from \(t=2\) to \(t=9\).
- On a position-time graph, the points for an interval are \((2,6)\) and \((8,0)\). What is the secant slope, and what does its sign indicate?
- A particle has the same position at the beginning and end of a time interval. What is its average velocity? What does that result not establish about its motion during the interval?
- Explain why the average velocity from positions measured in kilometers and times measured in hours has units of kilometers per hour.
