One way we apply derivatives to real life through rates of change involves position and motion. These problems involve rectilinear motion. 🚗
These problems always start with a particle, and that particle is moving in some way. You can think of position, x(t), as the original function at any given time. When you plug in time to the position function, you get out the location of the particle at that time.
That particle had to get to that position somehow! That is where velocity comes in. Velocity is always a measurement with respect to time, so it this is the derivative of our function with respect to time. You will see it written as v(t), but it may be helpful to picture it as x'(t), so you remember how to relate it to its position!
When velocity is negative, the particle is moving to the left. ➖ = ⬅️
When velocity is positive, the particle is moving to the right. ➕ = ➡️
If you want to talk about how fast the particle is moving but not about the direction, you are talking about speed. Speed is just velocity without direction. In math, the way we make direction not matter is by using absolute value. So we can think of speed as v(t)
What do you get when you find the rate of change of the velocity, you get how fast your velocity is changing, which is acceleration! Acceleration is the derivative of velocity, meaning it is the second derivative of position. a(t) = v'(t) = x''(t).
When v(t), a(t) have the same sign, then the particle is speeding up! ➕➕ = ⬆️
When v(t), a(t) have different signs, then the particle is slowing down! ➕➖ = ⬇️
When solving problems about rectilinear motion, you will often be given the position, velocity, or acceleration in any form (graph, function, table), and asked to find information about another measurement, that you may not have been given, at a specific time. You must be able to understand and move freely between position, velocity, and acceleration.
You may also be asked to analyze the motion of the particle. When this is asked, you need to find a few things. ❗
Find where v(t)= 0 and a(t) = 0
Use those values to figure out where the velocity and acceleration functions are negative and positive.
Use the velocity function to figure out what direction the particle is moving.
Use the acceleration and velocity to figure out where the function is speeding up and slowing down.
Use the values of t from where velocity and acceleration are zero and plug them into the position function to figure out where the particle was when it decided to change position.
This is a lot of information, so sign charts can help!
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4.2Straight-Line Motion: Connecting Position, Velocity, and Acceleration
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