# Finding when the velocity equals x.

#### Kakariki

Hey! I know I've had a lot of questions on here over the past couple days, but I am just ensuring I understand what I am doing, and you guys do such a great job of explaining everything!
Okay, I have a derivative question that I am not entirely sure how to solve, so I was hoping you guys could help me out!

Question
The position function of an object moving along a straight line is given by the function $$\displaystyle s(t) = t^3 - 15t^2 + 48t - 10$$, where s is in metres and t s in seconds and $$\displaystyle 0<t<15$$.
i) When is the velocity the object greater than 21 m/s?
ii) When is the speed of the object less than 21 m/s?

Question
I really have no idea how to solve for this! With the equation given I know the distance in metres after t seconds, but not the velocity, and the question is asking for velocity.
How would I go about solving this question?

#### pickslides

MHF Helper
i) When is the velocity the object greater than 21 m/s?
Find $$\displaystyle s'(t)>21$$

ii) When is the speed of the object less than 21 m/s?
Do you mean $$\displaystyle \frac{m}{s^2}$$ ?

This needs to be confirmed.

I think you are after $$\displaystyle s''(t)<21$$

#### p0oint

@pickslides
Speed is the rate of change of distance with time.
Velocity is the rate of change of displacement with time.

which means:

Speed is the first derivative of distance with respect to time.
Velocity is the first derivative of displacement with respect to time.

Let me clarify.

v(t)=s'(t), that is the instantaneous velocity at time t.

Velocity is speed with direction, and speed is velocity with no direction, so the speed equals |v(t)| or |s'(t)|.

What she needs to find is:

i) s'(t)>21

ii)|s'(t)|<21

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