53 lines
2.6 KiB
Markdown
53 lines
2.6 KiB
Markdown
---
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title: Vectors and Kinematics
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date: 2024-08-19
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---
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> One must always start a study into the heavily crippled IB editions of the glorious subject of Physics with the initial understanding that the road ahead lead to pain immeasurable.
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>
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> -> Prime of the Faith
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# Motion in 1D
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In this rendering of motion, you will *never* need to use vector quantities to describe movement due to the scalar nature of all quantities discussed.
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## Reference Frames and Displacement
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- Any measurement about motion is taken in terms of a *reference frame*.
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> [!NOTE]
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> **Example:** A train that moves with respect to the ground being held stationary, is not moving with respect to a stationary person inside that train. If a person were to walk, at let's say $5$ km/s toward the back of the train while it moves forward at $80$ km/h, the person is moving at $75$ km/h with respect to the stationary ground.
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- In one dimension, only one axis, the $x$-axis of a coordinate plane is used.
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- the *net* distance an object has traveled is known as *displacement*
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- *total distance* is the overall distance traveled by the object/particle regardless of reference frame or initial/final positions
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- change in positions is described using $\Delta x$
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## Average Velocity
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It is important to note that this equation was derived from the more complicated calculus variant of the velocity equation. There are 2 important terms here.
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$$
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\text{average speed}=\frac{\text{distance traveled}}{\text{time elapsed}}=\frac{\Delta x}{\Delta t}
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$$
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For any form of velocity, the speed is simply calculated by using the magnitude of the velocity vector $||\vec{v}||$, but since we exist in one dimension now, we will use $|v(t)|$
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## Instantaneous Velocity
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If the position equation is defined as $x(t)$, then:
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$$
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\text{instantaneous velocity} = \tfrac{\mathrm{d}x}{\mathrm{d}t} = v(t) = \lim_{\Delta t \to 0}{\tfrac{\Delta x}{\Delta t}}
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$$
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## Average Acceleration
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The acceleration of an object is the rate at which the velocity of said object changes. **Average Acceleration** is defined as the change in velocity from 2 distinct points divided by the change in time between those 2 distinct points.
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$$
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\text{average acceleration} = \frac{\text{change of velocity}}{\text{time elapsed}}
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$$
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Or more mathematically:
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$$
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a=\frac{v_2-v_1}{t_2-t_1}=\tfrac{\Delta v}{\Delta t}
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$$
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## Instantaneous Acceleration
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If the velocity function is defined as $v(t)$ (this notation only applies to 1 dimension):
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$$
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a(t)=\tfrac{\mathrm{d}v}{\mathrm{d}t}=\lim_{\Delta t \to 0}{\tfrac{\Delta v}{\Delta t}}
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$$
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Please take calculus/study calculus if you want a neuron or two to function during the course of this, well, course.
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#physics
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