Learning Outcomes

Upon completing this lesson, you will be able to:

• Analyze rectilinear motion when acceleration is a function of position or velocity.
• Construct velocity-time and acceleration-time graphs from a given position-time graph, and vice versa.
• Understand and apply the graphical relationships between position, velocity, and acceleration (slope and area).
• Interpret velocity-position ((v-s)) and acceleration-position ((a-s)) graphs to find kinematic quantities.

Topics Covered

• Acceleration as a Function of Velocity or Position
• The $s-t$, $v-t$, and $a-t$ Graphs
• The $v-s$ and $a-s$ Graphs

I. Acceleration as a Function of Velocity or Position

In many real-world scenarios, an object's acceleration is not constant, nor is it a simple function of time.

For example, the drag force on an object moving through a fluid depends on its velocity, meaning its acceleration is a function of velocity.

To handle these cases, we use the fundamental kinematic relationship derived from $v = ds/dt$ and $a = dv/dt$:

$$ a , ds = v , dv $$

This equation is essential for relating position, velocity, and acceleration when time is not explicitly part of the function.

A. Acceleration as a Function of Velocity $(a(v))$

If acceleration is given as a function of velocity, we can find the time or position by separating variables and integrating.

• To find time $(t)$:

  $$ a = dv/dt \implies dt = \frac{dv}{a(v)} \implies t = \int \frac{dv}{a(v)} $$

• To find position $(s)$:

  $$ a , ds = v , dv \implies ds = \frac{v , dv}{a(v)} \implies s = \int \frac{v , dv}{a(v)} $$

B. Acceleration as a Function of Position $(a(s))$

If acceleration is given as a function of position, we can find the velocity.

• To find velocity$(v)$:

  $$ a , ds = v , dv \implies \int a(s) , ds = \int v , dv\\\int_{s_0}^{s} a(s) , ds = \frac{1}{2}v^2 - \frac{1}{2}v_0^2 $$

II. The $s-t$, $v-t$, and $a-t$ Graphs

Erratic motion is often described using graphs. The relationships between the position-time $(s-t)$, velocity-time $(v-t)$, and acceleration-time $(a-t)$ graphs are derived directly from the definitions of velocity and acceleration.

A. The $s-t$ Graph

• The value on the $y$-axis gives the position at any time $t$

• The slope of the $s-t$ graph at any point gives the instantaneous velocity at that point:

  $$ (v = ds/dt) $$

  • A positive slope means positive velocity.
  • A zero slope (horizontal line) means zero velocity (at rest).
  • A curved line indicates that velocity is changing (i.e., there is acceleration).

B. The $v-t$ Graph

• The value on the $y$-axis gives the velocity at any time $t$

• The slope of the $v-t$ graph at any point gives the instantaneous acceleration at that point:

  $$ (a = dv/dt) $$

• The area under the $v-t$ graph between two times represents the change in position (displacement) during that interval:

  $$ (\Delta s = \int v , dt) $$

C. The $a-t$ Graph

• The value on the $y$-axis gives the acceleration at any time $t$

• The area under the $a-t$ graph between two times represents the change in velocity during that interval:

  $$ (\Delta v = \int a , dt) $$