⊕Read Tipler and Mosca (2004) chapter 2 (Motion in 1D); Pelcovits and Farkas (2024) chapter 2 (Kinematics). See also https://www.youtube.com/watch?v=ZM8ECpBuQYE

Recall that position, velocity, and acceleration are related; velocity is the derivative of position, and acceleration is the derivative of velocity. We will use this to examine two special cases of motion in 1D: motion at constant velocity, and motion at constant acceleration.

For simplicity, we will drop the vector notation here because we are restricting our discussion to one dimension. We’ll return to vectors when we generalize these results to motion in two- and three-dimensions next time.

1D motion with constant velocity

Consider a C-130 rolling down the strip. Big big bird, gonna take a little trip. As a hello world introduction to equations of motion, we will assume it is moving at constant velocity \(V\), so that the acceleration \(a\) is 0. What is the position for this case? Back in middle school we would have done something like distance equals speed times time, and here we would have \(x = V t\), though perhaps we might adjust for a starting point by adding an \(x_0\) initial position term. The equations of motion are then:

\[\begin{aligned} x(t) &= x_0 + V t \\ v(t) &= V\ \text{(constant)} \\ a(t) &= 0 \end{aligned}\]

Position \(x(t)\) is linear; its intercept is \(x_0\) and its slope is \(V\) which is also the velocity. Velocity \(v(t)\) is just a constant; while acceleration \(a(t)\) is zero. These three together are equations of motion. They are parametric in \(t\) and give a fairly complete view of the movement of the object.

We can also check these and see that our derivative relationships are satisfied too, as \[\dfrac{dx}{dt} = V = v(t),\] and \[\dfrac{dv}{dt} = 0 = a(t).\]

The equations hold for 1D motion at constant (linear) velocity, which means the speed and direction of the object are not changing.

1D motion with constant acceleration

⊕See https://www.youtube.com/watch?v=ZBr8Q2ROX9s&t=1s

We could develop this assuming \(a\) is a constant and integrating twice but you do not yet know how to integrate. Assuming \(a\) is a constant, if the velocity were of a form \[v(t) = v_0 + a t\] then the derivative of velocity would just be

\[\begin{aligned} \dfrac{dv}{dt} &= 0 + a \\ &= a\ \text{(constant)} \\ &= a(t). \end{aligned}\] So far, so good.

Can we think of a function \(x(t)\) where its derivative gives us \(v(t)\)? \[x(t) = x_0 + v_0 t + \frac{1}{2} a t^2\]

If we take the derivative of \(x(t)\), then we get \[\begin{aligned} \dfrac{dx}{dt} &= 0 + v_0 + 2 \frac{1}{2} a t \\ &= v_0 + a t \\ &= v(t). \end{aligned}\] Done!

Putting these together, the following equations hold for 1D motion with constant (linear) acceleration, which means there is a constant net force acting on the object that makes it go faster or slower.

\[ x(t) = x_0 + v_0 t + \dfrac{1}{2} a t^2 \qquad{(1)}\] \[ v(t) = v_0 + a t \qquad{(2)}\] \[ a(t) = \text{(constant)} \]

eq. 2 and eq. 1 are the first two equations provided for you on your AP Physics C Mechanics equation sheetSee https://apcentral.collegeboard.org/media/pdf/ap-physics-c-mechanics-equations-sheet.pdf

, so you’d better know how to use them.

Additional equations

There are two additional equations that come up, often in the context of non-calculus based physics classes (e.g. AP Physics 1).

The first is also provided for you as the third equation on your AP Physics C Mechanics equation sheetThis equation I usually think of as coming from energy conservation, where the final kinetic energy is equal to the initial kinetic energy plus the work done in accelerating the object, \(\frac{1}{2}mv^2 = \frac{1}{2}mv_0^2+ma(x-x_0)\).

: \[v^2 = v_0^2 + 2 a (x - x_0)\] which gives the final velocity as a function of the initial velocity, the acceleration, and the displacement. It can be useful for doing certain types of problems where what is needed is the final velocity, or the distance needed to get up to a certain velocity.

The other equation that shows up is the “average” speed. Assuming the velocity is linearly increasing, the average speed is given by \[v_{av} = \dfrac{v_0 + v_f}{2}.\qquad{(3)}\]

Eq. 3 only works for linearly increasing velocity. Otherwise, or if in doubt, take the average speed as \[v_{av} = \dfrac{x_f - x_0}{\Delta t}\] or \[v_{av} = \dfrac{1}{t_f - t_0} \int_{t_0}^{t_f} v(\tau) d\tau.\]

Frames of reference and relative velocity (1D)

A brief aside on frames of reference and relative velocity. Imagine a moving walkway at an airport that has constant velocity \(v_w\). A person gets on the walkway and they are walking on it at \(v_p\), which is their velocity, relative to the walkway. As a result, an observer that is stationary would see the person moving at \(v_w+v_p\). The velocity of the person relative to the walkway, \(v_p\), is called relative velocity. The person on the walkway is in a moving frame of reference.

In general, to an observer at rest, the velocity of something moving \(\vec{v}_{rel}\) relative to a inertial frame of reference moving at \(\vec{v}_{frame}\) is the sum: \[\vec{v}_{rest} = \vec{v}_{frame} + \vec{v}_{rel}.\]

When a frame of reference is not accelerating, we call it an inertial reference frameWhen a frame of reference is accelerating, it is a non-inertial reference frame. The acceleration and movement of the reference frames can lead observers within it to see funny acclerations and pseudoforces. And example of this might be the sideways “force” felt when you are in a car moving along a circular turn. We will discuss forces more later.

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When velocities are near the speed of light, things get weirdAs velocities get closer to the speed of light, funny things also happen. The transformations between frames becomes more complicated based on the idea that nothing should go faster than light; this results in some differences between what stationary and moving observers see. You will cover this more when we discuss relativity but this is well beyond the scope of AP Physics C Mechanics.

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See also

• https://www.youtube.com/watch?v=ZM8ECpBuQYE

• https://www.youtube.com/watch?v=livG1fp-q8s

• https://www.youtube.com/watch?v=szCChCc58dg

• Red Bull demonstration of relative velocity https://www.youtube.com/watch?v=TP_0Vv5F29I

• Trampoline on a moving cart https://www.youtube.com/shorts/yxOunOndh4Y

• Jim Al-Khalil recreation of Galileo inclined plane experiment https://www.youtube.com/watch?v=ZBr8Q2ROX9s&t=1s

Navigation links

• Previous: Position, velocity, and acceleration
• Table of contents
• Next: Motion in 2D

References