How do you calculate velocity and acceleration?

Definition

Velocity is the rate of change of position with respect to time. It is a vector quantity, meaning it has both magnitude and direction. Mathematically, if $s(t)$ is the position at time $t$, then velocity $v(t)$ is:

$$ v(t) = \frac{ds}{dt} $$

Acceleration is the rate of change of velocity with respect to time. It is also a vector quantity. If $v(t)$ is the velocity at time $t$, then acceleration $a(t)$ is:

$$ a(t) = \frac{dv}{dt} = \frac{d^2s}{dt^2} $$

Worked Example

Suppose the position of a particle is given by:

$$ s(t) = 3t^2 + 2t + 1 $$

where $s$ is in meters and $t$ is in seconds.

Step 1: Find the velocity

Take the first derivative of $s(t)$ with respect to $t$:

$$ v(t) = \frac{ds}{dt} = \frac{d}{dt}(3t^2 + 2t + 1) = 6t + 2 $$

Step 2: Find the acceleration

Take the derivative of $v(t)$ with respect to $t$:

$$ a(t) = \frac{dv}{dt} = \frac{d}{dt}(6t + 2) = 6 $$

So, for this example:

• Velocity at time $t$ is $v(t) = 6t + 2$ (m/s)
• Acceleration is constant: $a(t) = 6$ (m/s$^2$)

Takeaways

• Velocity is the first derivative of position with respect to time; acceleration is the derivative of velocity.
• For $s(t) = 3t^2 + 2t + 1$, velocity is $6t + 2$ and acceleration is $6$.
• Calculating these involves basic differentiation.