Coaching Centre
Instantaneous velocity and speed
Velocity as the derivative of displacement, speed as its size, vectors and scalars, and finding when a particle is stationary.
Velocity as a derivative
- $v > 0$: moving in the positive direction. $v < 0$: moving in the negative direction.
- Speed is $|v|$, the size of the velocity.
- The particle is stationary when $v = 0$.
- Displacement and velocity are vectors (they have direction); distance and speed are scalars.
A particle has displacement $x = t^3 - 6t^2 + 9t$. Find when it is stationary, where it is then, and its speed at $t = 2$.
Orange: displacement x. Blue dashed: velocity v. Where v crosses zero, the particle turns around.
Between which times is the particle moving in the negative direction?
A ball has height $h = 20t - 5t^2$ metres. Find its greatest height.
For $x = t^3 - 6t^2 + 9t$, find the velocity at $t = 0$.
$v = 3t^2 - 12t + 9$.
When is that particle stationary? Enter both times.
$3(t - 1)(t - 3) = 0$.
Where is it the first time it is stationary?
$x(1) = 1 - 6 + 9 = 4$.
Find its speed at $t = 2$.
$v = -3$, so the speed is $3$.
A negative velocity means the particle is:
The sign of $v$ gives the direction of motion, not the position.
A ball has height $h = 20t - 5t^2$. Find its velocity at $t = 1$.
$20 - 10 = 10$ m/s.
When does the ball reach its greatest height?
$20 - 10t = 0$.
Find the greatest height, in metres.
$h(2) = 20$.
Which pair are both scalars?
Scalars have size only.
New numbers every time. Keep going until you get five in a row.