How to Find Average Velocity of a Multi-Part Trip

A car problem: find v (preview)

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Solve for v: 80v/(v + 60) = 16. Multiply both sides by v plus 60: 80v equals 16v plus 60 times 16, which is 960. So v equals 15 m/s. So for the car's second segment, the average velocity is 15 m/s. Choose C. Today: multi-segment average velocity

Two big ideas: piecewise vs. overall analysis

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This type comes up a lot in exams, often tested on velocity in the Describing Motion chapter. First, a quick aside: the big-picture way to approach high school physics problems. Listen up: for any physics problem, there are only two big ways to analyze it. The first is piecewise analysis. Say you get a complex motion and need the total time or distance. You can split it into many segments: solve the first, second, third. Even if the whole process is complicated, each segment is usually simple to solve. Solve each, then add them up. Split first, then combine. That's piecewise analysis. The second way is overall analysis. For some problems, the middle may be very complicated, but you can ignore the changes in between and analyze the whole directly, which is often simpler.

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Take an object thrown straight up, which we'll cover soon. You throw a small stone straight up. It's vertical projectile motion. Two stages: slowing down as it rises to zero, then speeding up as it falls. Piecewise analysis is messier; one equation for the whole is simpler. Later, in force analysis, you'll learn the whole-system method. When you stack a bunch of things and analyze the forces, treat them as one whole and ignore the forces between them, and the analysis gets simpler. We'll also learn a key tool: the work-energy theorem. The middle can be very messy: steady, then speeding up, then slowing down. A hassle. But looking only at the start and the end, write one equation for the whole trip, the work-energy theorem, ignoring how it moves between.

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One equation solves it. These are all examples of overall analysis. Both methods are important. Most problems use piecewise analysis, but a few problems work wonders with overall analysis. Don't worry, in the later lessons I'll go through them one by one. Today's multi-segment average velocity problem is a classic case for overall analysis. It's the first type in high school physics to use overall analysis. In this lesson, I hope you'll train with me to see problems as a whole. That matters even more than learning how to solve this one type of problem. Practice overall analysis. Now, the features of this problem type:

Problem features: one-way straight-line motion

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They give one-way straight-line motion, split into several segments, with information about each. They test if you can use overall analysis to find the whole-trip average velocity. So all of today's problems are one-way straight-line motion. By how the motion is split, there are two kinds. Type one: the whole trip is split by time. Look at this problem: I split it into the first two-thirds of t and the last one-third of t. I know the average velocity in each segment and need the whole-trip average velocity. Type two: we split by displacement. In this problem, the trip is split into the first one-third of the displacement and the last two-thirds. I again know each segment's average velocity and need the whole-trip average velocity.

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When analyzing this type,

Core idea: total displacement ÷ total time

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keep telling yourself: overall, overall, overall, overall. I look for the whole. I analyze the whole. How is average velocity defined? I don't care if it's uniform or changing, speeding up, slowing down, or both. To find this motion's average velocity, I need its total displacement, total time. So put all your attention on finding the total displacement and the total time. That's the overall way of analyzing. First, find the total displacement. Second, find the total time. Third, just divide. Once more: focus on the whole. We look for total displacement, and total time. That's our goal. Next, problems split by time. A problem split by time already tells you

Split by time · Example 1

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the time information you need. So the key: use time to get total displacement. Remember, it's still overall analysis: we want total displacement and total time. Split by time? Find total displacement. Let's work it out. Take this problem: a particle moves in a straight line, always one way. One-way motion, right? First 2/3 of the time: average velocity v/2. In the last 1/3 of the time, it's 2v. Find the whole trip's average velocity. Let total time be t. Segment one is 2t/3, and segment two is t/3. We find each segment's displacement, then add them up. First segment: time is 2t/3, average velocity is v/2, so x1 equals two-thirds of t times v over 2, which is vt over 3.

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Second segment: x2 is one-third of t times 2v, its average velocity. That works out to two-thirds of vt. Add them for total displacement. Total displacement equals x1 plus x2, which is vt. Add them up: vt. Average velocity is total displacement divided by total time, which is vt divided by t. Isn't that just v? So the answer is C. Use this problem to practice overall analysis. What we want in the end is total displacement and total time, both overall quantities. Concretely, we find total displacement. This problem gives time information, so we use time to find it. It's not hard; get a feel for overall analysis.

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Now another one, split by displacement. Split by displacement means

Split by displacement · Example 2

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you're given displacement information, so the key is using it to find total time. Here's the problem, intro left out: one-way straight-line motion. In the first 1/3 of the displacement, average velocity is v1. In the last 2/3, it's v2. Find the whole trip's average velocity. It's split into two segments by displacement. Let the total displacement be x. The first segment is x over 3, the second is 2x over 3. So the first segment's time, t1, equals x over 3 divided by v1. That's the first displacement divided by the first average velocity: x over 3v1. The second segment's time is 2x over 3, divided by v2, which is 2x over 3v2. The total time is t1 plus t2.

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The whole trip's average velocity is total displacement over (t1 + t2). Plugging in: x over (t1 + t2). This takes more calculation, doesn't it? x over [x/(3v1) + 2x/(3v2)]. First, the x's cancel out. It's all gone. x is gone. Now simplify the denominator first. It becomes (v2 + 2v1)/(3v1v2). Multiply 1/(3v1) top and bottom by v2, do the same to the other with v1, then add. That's just math, so let's work it out. Flip it over: 3v1v2/(v2 + 2v1). Splitting by displacement like this makes the calculation a bit nasty. The result: 3v1v2/(v2 + 2v1). That's the displacement-split approach: use displacement to find time.

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In short: split by time, use time to find displacement. Split by displacement, use displacement to find time. Last problem of this lesson. It's a minor type:

Example 3: find one segment from the whole trip

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Do three to practice overall analysis. No need to do many. A car drives along a straight road. At first, 20 m/s for 1/4 of the distance, Then speed v for the remaining 3/4. The whole-trip average velocity is 16 m/s. Find v. First, it's split by displacement: one quarter and three quarters. This one differs from the earlier ones: it gives the whole-trip average velocity and asks for one segment. But we analyze it the same way. Since it's split by displacement, use displacement to find time. Let the total displacement be x. Then the first segment's time, t1, equals the displacement x/4, divided by its average velocity, 20. That works out to x over 80.

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Second segment: three quarters at speed v. So t2 equals three quarters of x, over v, which works out to 3x over 4v. How do we get whole-trip average velocity? It equals x divided by (t1 + t2). That is x divided by (x/80 + 3x/(4v)). That's the sum of the two times. Simplify it yourself; the x cancels. It comes out to 80v/(v + 60). Right? Does it equal 16? Yes: the whole-trip average velocity is 16. Look at this equation: v is the only unknown, so we can solve for v. Solve for v: 80v/(v + 60) = 16. Multiply both sides by v plus 60: 80v equals 16v plus 60 times 16, which is 960.

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So v equals 15 m/s. So for the car's second segment, the average velocity is 15 m/s. Choose C.

Recap: split by time or by displacement

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These problems give the segments and ask for the whole trip, or give the whole trip and ask for a segment. Either way, the core idea is the same. Check if it's split by displacement or split by time. Time split: use time to find displacement. Displacement split: use it to find time. In the end, it all comes down to: whole-trip average velocity equals total displacement over total time. That's how overall analysis works. We covered multi-segment average velocity

Summary

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Its feature: one-way straight-line motion. Sometimes the problem doesn't say so. But for multi-segment problems, it's one-way straight-line motion. Don't worry: exams always set it up this way. Problems aren't random; they follow patterns. It gives each segment's average velocity, split either by time or by displacement, and asks for the whole trip, or gives the whole trip and asks for a segment. Same thing, really. The core idea: find the total displacement and the total time, then divide. If it's split by time, use time to find displacement. If it's split by displacement, use displacement to find time.