Calculus kinematics
Enrichment article · Physics C: Mechanics
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The finite difference in source paragraph p10 is an average rate. A derivative is the limit of these average rates as the time interval approaches zero, where that limit exists; keep that distinction when reading the historical wording below.
In actuality, the values of position, velocity, and acceleration are very closely related, and all are related through time. To make this more obvious, we can look at the units of each.
Average velocity is displacement divided by elapsed time. Average acceleration is velocity change divided by elapsed time. Instantaneous velocity and acceleration are derivatives.
Average velocity is v_avg = Δx/Δt (Equation 3.1.1), and average acceleration is a_avg = Δv/Δt (Equation 3.1.2). Both definitions apply over any interval with nonzero elapsed time. Instantaneous values are obtained from derivatives.
The average formulas hold over any nonzero time interval even when velocity or acceleration varies. They need not equal the instantaneous values at a particular time.
This concept relates to average velocity and acceleration. Basically, the average velocity of some particle is given by the above equation, for any interval . This doesn’t mean, however, that the velocity at every point is equal to that average velocity. That concept we will discuss later.
First things first, we must understand the relationships between average velocity and acceleration. To identify the average velocity given any known position history, be it a word problem or a graph, you simply use the previous equation. You just figure out the change in over the change in time.
That is actually a basic process in calculus called the derivative.
The ’s in those equations represent differentials, which refers to minute changes. The approach that we use to solve physics problems with this idea is only one way calculus students utilize derivatives, instead of taking derivatives of functions, we just use the basic idea of the changes in our system.
What you saw before, for the work in section 2.4, was actually an integral. Those are just the opposite of derivatives. Now, you can’t just take the opposite of the change, so we need an easier approach to dealing with those.
Now if you want to figure out what the velocity or acceleration is at a certain time, however, that requires a bit more skill.
That is what you saw with regards to the area under the curve.
Specifically the relationships in the integrals are given by the following formulas:
v(t₂) − v(t₁) = ∫ from t₁ to t₂ of a(t) dt; x(t₂) − x(t₁) = ∫ from t₁ to t₂ of v(t) dt.
Signed area under a velocity–time graph gives displacement. Total distance is the area under the speed graph |v(t)|.
Or, if you have a graph of acceleration, and you want to know how much the velocity changes, you find the area under the curve.
The tangent slope of a velocity–time graph gives instantaneous acceleration. A secant slope gives average acceleration over an interval.
Instantaneous velocity is dx/dt and instantaneous acceleration is dv/dt, the limits of the corresponding average rates as the interval approaches zero.
For a graph, the tangent slope gives the instantaneous rate wherever that derivative exists; a straight segment has a constant slope.
The technique I am referring to is the slope-analysis. This technique is only possible when you have a graph of either position versus time, if you want to solve for velocity, or velocity versus time, if you want to solve for acceleration.
Slope-analysis is simply figuring out the slope of the graph at the point in question.
Source and curriculum context
Adapted from Physics I by Jake Yap and Laurence Vincent.
physics-c-mechanics · 1.2.C.1.i, 1.2.C.1.ii, 1.2.C.2 · printed pages 30.
Glossary and deeper reading · 5 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Velocity
Velocity describes how position changes with time and includes direction. Instantaneous velocity describes one moment, while average velocity is displacement divided by elapsed time. For example, a car turning a corner can change velocity while maintaining the same speed.
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Acceleration
Acceleration is the vector rate of change of velocity with time. It can change an object’s speed, direction, or both, and it is measured in meters per second squared. For example, a satellite moving at constant speed in a circular orbit accelerates toward the center.
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Average velocity
Average velocity is the displacement over an interval divided by the elapsed time. It is a vector and need not equal the average speed, which uses total distance. For example, a runner who returns to the starting line has zero average velocity for the full lap.
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Displacement
Displacement is the vector change in position from the start of an interval to its end. It depends on those endpoints rather than the length or shape of the path traveled. For example, walking around a closed track and returning to the start gives zero displacement but a nonzero distance traveled.
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Instantaneous velocity
Instantaneous velocity is the rate of change of position at a particular moment. On a position-versus-time graph, its component is the slope of the tangent at that time. For example, the velocity just before a ball lands differs from the average velocity of its entire flight.