Displacement, Velocity, and Acceleration


Now that we have an understanding of dimensions, units, vectors, and scalars let’s move on to exploring some of them. Since we are talking about motion we will be looking at distance and time. After all, motion occurs when we move over some amount of space and that takes time (t).

Distance Versus Displacement

First, we will explore the idea of distance. Distance (d) is a running total of how far we traveled measured in the SI units of meters (m). This is the number that a GPS spits out when planning a trip to the museum and back making it a scalar quantity.

Distance for a one way trip is shown on the right while the distance for the round trip is shown on the left.

Displacement (x) on the other hand is a vector quantity also measured in meters (m). This means that all that matters is where we start and end, not how we get there. So even though we went from home to the museum we haven’t actually displaced ourselves since we ended right where we started.

Displacement for the one way trip shown on the left while displacement for the round trip is shown on the right. We will always draw a straight arrow from start to finish for displacement whereas distance traced the actual path we took to the museum.

Speed Versus Velocity

The contrast between speed and velocity is the same as that between distance and displacement. Speed, measured in the SI unit m/s, tells you how fast you cover a specific distance. For example, when you got on the highway to go to the museum you were traveling at a specific speed, let’s say 30m/s. This means that you traveled 30 meters in one-second while on the highway.

In contrast velocity (m/s) tells you how fast you change your position. So if we looked at our whole trip from home to the museum and back you would have an average velocity of zero. No displacement, no position change, and no average velocity.

Hey, wait a second not so fast. Didn’t we have a velocity at some point throughout our journey? Of course, we did! So long as we have some displacement we have a velocity the key is what perspective we choose to take.

If we look only at our trip to the museum or only our trip back home then we had displacement and velocity. However, if we look at the journey as a whole our displacement is zero and thus our velocity is also zero. This is simply because they are vector quantities and we have to add them together tip to tail as we did before. When we do this they cancel leaving us with zero velocity.

However when we consider just the trip from our house to the museum a resultant vector remains as seen above. Here all the matters is our starting point and our ending point though so the displacement travelled is a lot less than the distance. As a result our average speed would end up being greater than our average velocity.

To illustrate this let’s say it took 30 minutes to go from our house to the museum. The total distance covered in this time is 10 miles while the total displacement is 3 miles. This means that the average speed throughout was 0.33 miles/min* while the average velocity was only 0.1 miles/min.

[latexpage]
\[
Speed=\frac{\delta d}{\delta t}\to \frac{10\; miles}{30\min} \to 0.33\frac{miles}{min}
\]

\[
Velocity=\frac{\delta x}{\delta t}\to \frac{3\; miles}{30\min} \to 0.1\frac{miles}{min}
\]

*Non-SI units are used here for comparison purposes only if we need a speed or a velocity for another equation we would need to convert this to meters per second.

Instantaneous or Average

Another lens to view both velocity and speed is in how we measure them. For example, we could look at our velocity on the highway or in our neighborhood. These instances have different velocities and describe specific moments in time we would call this an instantaneous velocity.

On the other hand, we could look at our overall velocity for our trip from home to the museum. We would take all of the instantaneous velocities and average them together. Or we could just take the overall displacement we traveled and divide it by the time it took to travel to the museum giving us an average velocity.

[latexpage]
\[
v_{avg.}=\frac{x}{t}
\]
or in units
\[
v_{avg.}=\frac{m}{s}
\]

Average velocity is typically more useful since it can be manipulated and used in calculations on the MCAT although the distinction between the two isn’t particularly important.

Acceleration

In order to reach highway speeds while heading to the museum, we had to speed up. Acceleration (a) is the measure of how quickly we did this and measures our change in velocity over time. Since it is based on velocity it is also a vector quantity, must have a direction as well, and has the units of m/s2.

[latexpage]
\[
a=\frac{\delta v}{\delta t}
\]

Just as before we had a net-zero acceleration if we view our trip as a whole. However, if we only look at one leg of the journey then we had a net acceleration.

Sign Conventions

By and large, physicists aren’t drawing pictures and arrows all the time to denote what direction all of these vectors are going. Instead, physicists tend to use sign conventions to describe which way we are moving. In motion and forces, these are based on the standard x-y coordinate plane.

So if we move to the right or up we will have positive displacement, velocity, and acceleration. However, if we move the left or down our displacement, velocity, or acceleration will be negative. Generally speaking the MCAT doesn’t discuss specific x,y coordinates but will instead use language to convey sign.

There are two major ways this shows up in the MCAT.

  1. They will define a positive vector and we must determine if the answer is in the same (indicates positive answer) or opposite direction (indicates a negative answer)
  2. They will use the term acceleration to indicate positive acceleration and deceleration to indicate negative acceleration

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