Calculus / Unit 5: Applications of Derivatives I
5.1 Position, Velocity and Acceleration
Notes
For an object moving along a line with position $s(t)$:
- Velocity $v(t) = s'(t)$ (positive means moving in the positive direction, negative means the opposite).
- Acceleration $a(t) = v'(t) = s''(t)$.
- Speed is $|v(t)|$.
- Example: $s(t) = t^2 + 3t$ gives $v(t) = 2t + 3$ and $a(t) = 2$, so $v(2) = 7$.
Practice
A particle has position $s(t) = t^2 + 3t$. What is its velocity at $t = 2$?
$v(t) = 2t + 3$, so $v(2) = 7$.
A particle has $s(t) = 2t^3$. What is its acceleration at $t = 1$?
$v = 6t^2$ and $a = 12t$. So $a(1) = 12$.
Which units fit acceleration if position is in meters and time is in seconds?
Acceleration is the rate of change of velocity, so units are meters per second per second.
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