Classical Mechanics
# Kinematics

In the last quiz, we were able to relate position, velocity, and acceleration in special cases. We were also able to write down the relationship between the pairs \((a,v)\) and \((v,d)\).

In this quiz we're going to formalize our findings, **derive the general kinematic relations**, and obtain the relations for the special case of constant acceleration, \(a(t) = a_0\).

When we explored **Kinematics in the City** we saw that for constant velocity, \(d=vT\).
In general, when \(v\) depends on time, this relation is a **differential equation** for the rate of change of displacement \(d\).
How can we write this as a differential equation?

Suppose that we break up the \(v\) vs. \(t\) plot into rectangles. We can write \[r_n = v_1\Delta t + v_2\Delta t + \cdots + v_n \Delta t.\] What is \(r_n - r_{n-1}\)?

Suppose you're given the following form for the acceleration of a particle that starts from rest: \[a(t) = a_0 e^t.\]

Find the velocity at time \(t_f,\) \(v(t_f),\) given that the initial velocity, \(v(t_i),\) is zero.

Suppose we're given an arbitrary form for the acceleration \(a(t)\) of a particle that starts from rest at \(r(t_i) = 0\).

Find its position \(r(t_f)\).

This is all a bit abstract. Let's apply our formula to a case that we've solved already. Recall our second motorcycle (the one that could briefly boost from \(v\) to \(v+\Delta v\)).

If it starts from rest at \(r(0)\), we can set \(v(0)\) equal to zero, and write the acceleration \(a(t)\) as \(a_0\).

Then, \[\begin{align} r(t_f) &= r(0) + \int\limits_0^{t_f} dt \int\limits_0^{t} dt^\prime a_0 \\&= \int\limits_0^{t_f} dt\ a_0 t \\ &= \frac12 a_0 {t_f}^2. \end{align}\]

Putting it all together, if our motorcycle has an initial velocity \(v(0) = v_0\), then we find that the position is given by

\[\begin{align} r(T) &= r_0 + \int\limits_0^T v_0 dt + \int\limits_0^T dt \left[\int\limits_0^t dt^\prime\ a_0\right] \\ &= r_0 + v_0T + \int\limits_0^T dt\ a_0 t \\ &= r_0 + v_0T + \frac12 a_0T^2. \end{align}\]

Suppose our motorcyclist starts at their apartment, with initial speed \(v_0\) and accelerates at a constant rate so that they're moving at \(v_f\) a time \(T\) later.

How far do they travel during this motion?

Sometimes it is necessary to relate the initial and final velocities of an object in terms of its distance traveled, without reference to the time. As one example, an insurance specialist may come to the scene of a motorcycle crash and know the braking deceleration of the motorcycle along with the distance of its skid, but want to know how fast it was going before the accident.

Suppose a motorcycle starts at \(v=v_0\) and then skids over a distance of \(d\), ending the skid at \(v=v_f\) .

Find \({v_f}^2 - {v_0}^2\) in terms of the distance \(d\) and the deceleration \(a\).

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