Mechanics

calculus-based physics for scientists and engineers

The Acceleration Function

The Acceleration Function in One Dimension

Are you practiced with the whole derivative thing? Good, because here we go again.

The acceleration function is the time-derivative of the velocity function.

Again, while there are many ways to write this mathematically, the fundamental relationship between the acceleration and the velocity is best to remember in words. Once you know this definition, regardless of the form of the velocity function, you can find the acceleration by taking the derivative with respect to time.

The average acceleration is defined the same way the average velocity was defined. Take two points in time, $$t_i$$ and $$t_f$$, where $$\Delta t = t_f- t_i$$. Find the velocities at those two times, $$v(t_i)$$ and $$v(t_f)$$, where $$\Delta v = v(t_f)- v(t_i)$$. The average acceleration in one dimension is given by: $$a_{avg} = \frac{\Delta v}{\Delta t}$$

Acceleration, Velocity, and Position Graphs

Let's look at the graphcial representation of the acceleration, velocity, and position all together. This gives us added insight in how the three quanties are related and different.

A Model of Falling Things

Make the following four assumptions.

1. The earth is flat.
2. The earth is not rotating.
3. There is no air to provide resistance to things falling.
4. Things are not very high (relative to the radius of the earth, which is over 6300 km).

A very good model of falling things is to assume everything -- everything -- falls directly toward the surface of the earth with a constant acceleration of 9.8 m/s2. Every falling thing has exactly the same acceleration: a vector with a magnitude of 9.8 m/s2 and a direction pointing down, perpendicular to the surface.

We use this number a lot, so we give the magnitude of this vector its own symbol: $$g=9.8$$ m/s2 Remember, this symbol, $$g$$ is the magnitude of the acceleration vector, so it is always positive.

Solution

1. A position function is given by $$x(t) = 4t - t^2$$, where position is in meters and time is in seconds. Find the velocity and acceleration functions. Sketch all three graphs, one on top of each other as in the video, with the same time axis for $$t_i =$$0 through $$t_f = 4$$.