Kinematics - Unit 1

University:
California State University, Chico
Course:
PHYS 202A | General Physics I
Academic year:

2022
Views:

397

Pages:

4
Author:

Maverick Haynes

Kinematics - Unit 1 There are 3 key quantities in kinematics and motion which we are interested in: position, velocity, and acceleration. This first unit only concerns motion in one dimension. The Kinematic Equations There are 3 main kinematic equations that you will need to use in the AP Physics course and exam. 1 𝑥 = 𝑥0 + 𝑣𝑥0 𝑡 + 𝑎𝑥 𝑡 2 2 𝑣𝑥 = 𝑣𝑥0 + 𝑎𝑥 𝑡 𝑣𝑥2 = 𝑣𝑥20 + 2𝑎𝑥 (𝑥 − 𝑥0 ) Constant/Average Velocity and Acceleration Models If velocity is constant, then velocity is given by the displacement and time elapsed. 𝑣𝑥 = 𝛥𝑥 𝛥𝑡 There are also similar models for average velocity and average/constant acceleration: 𝑣𝑥(𝑎𝑣𝑔) = 𝛥𝑥 𝛥𝑡 𝑎𝑥(𝑎𝑣𝑔) = 𝛥𝑣𝑥 𝛥𝑡 Calculus-Based Relationships and Models Now: the above kinematic equations only work in very, very specific situations: when acceleration is constant, and velocity/acceleration, etc. relationship isn’t changing with time, position, or some other variable. When that happens, you’ll need calculus-based relationships. If you have functions for velocity, position, or acceleration, then you need these models to relate each quantity. 𝑣𝑥 = 𝑑𝑥 𝑑𝑡 𝑎𝑥 = 𝑑𝑣𝑥 𝑑𝑡 We can extend these using the fundamental theorem of calculus: 𝛥𝑥 = ∫ 𝑣𝑥𝑑𝑡 𝛥𝑣 = ∫ 𝑎𝑥𝑑𝑡 If you took AP Physics 1, your teacher may have said something about the “area under the curve” on a velocity-time or acceleration-time graph: these integrals represent that. Acceleration and velocity can be nonuniform, and can follow any sort of function, really. Examples: Free-Fall Let’s imagine that you drop a basketball off a helicopter that is 500 meters above the ground on Earth. How long would it take to hit the ground? Let’s think: - We know that it will start with no initial velocity (you are dropping it after holding it still) - We know that it will have an acceleration downwards of g (which we can assume is 10 m/s2 downwards in AP Physics-land) - We know that it has to fall 50 meters to hit the ground - We DON’T know how long it will take, and we want to figure that out. Therefore, we want to use a kinematic equation that takes these variables into account. We will use the first one. 1 𝑥 = 𝑥0 + 𝑣𝑥0 𝑡 + 𝑎𝑥 𝑡 2 2 x0 is 0 m (we take the helicopter as the start point), x = 500m, vx = 0 m/s, and a = 10 m/s2. 1 2𝑥 𝑥 = 𝑎𝑥 𝑡 2 ⟶ 𝑡 = √ 2 𝑎 𝑡=√ 2(500) ⟶ 𝑡 = √100 = 10𝑠 10 Calculus-Based Example: Non-Uniform Velocity Imagine that we are doing calculations with a very fast experimental vehicle. When it goes full throttle, its velocity can be described with relation to time as 𝑣(𝑡) = 2𝑡 3 . Starting from rest, if the test driver floors the accelerator, how far would the vehicle travel in 5 seconds? Let’s think: velocity isn’t constant, so we can’t just multiply by time. However: we have a function for velocity that is with respect to time, and so we can just use a calculus-based relationship. Recall: 𝑡2 𝛥𝑥 = ∫ 𝑡1 𝑣(𝑡)𝑑𝑡 t1 is 0, and t2 is 5 (we start from 0 and go for 5 seconds). Let’s integrate! 5 𝛥𝑥 = ∫ 0 2𝑡3 𝑑𝑡 5 1 4 𝛥𝑥 = ( 𝑡 ) 2 0 𝛥𝑥 = 𝛥𝑥 1 4 (5 ) 2 = 312.5 𝑚𝑒𝑡𝑒𝑟𝑠