01_Motion_in_1D

1. Fundamental Concepts & Definitions

Point Particle (质点)
Displacement (位移): \(\Delta x\)
Average Velocity:

\[ v_{avg} = \frac{x_2 - x_1}{t_2 - t_1} \]
Average Speed:

\[ s_{avg} = \frac{\text{total distance}}{t_2 - t_1} \]
Instantaneous Velocity:

\[ v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt} \]
Instantaneous Speed:

\[ |v| \ge 0 \]
Average Acceleration:

\[ a_{avg} = \frac{v_2 - v_1}{t_2 - t_1} \]
Instantaneous Acceleration:

\[ a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{dv}{dt} = \frac{d^2x}{dt^2} \]
Dimension (量纲)

For a given \(a(t)\):

\[ dv = a(t)dt \]
Integrating both sides:

\[ v(t) = \int_{t_i}^t a(t') dt' + C_1 \]
Applying the initial condition \(v(t_i) = v_i\) (where "\(i\)" stands for initial):

\[ C_1 = v_i \]
Final Velocity Equation:

\[ v(t) = \int_{t_i}^t a(t') dt' + v_i \]

For a given \(v(t)\):

\[ x(t) = \int_{t_i}^t v(t') dt' + x_i \]
Substituting the expression for \(v(t')\) derived in Part 2:

\[ x(t) = \int_{t_i}^t dt' \left[ \int_{t_i}^{t'} a(t'') dt'' + v_i \right] + x_i \]
Expanding the integral gives the Final Position Equation:

\[ x(t) = x_i + (t - t_i)v_i + \int_{t_i}^t dt' \int_{t_i}^{t'} dt'' a(t'') \]

This section applies the general integral formulas to the specific case where acceleration is constant (\(a(t) = a\)).

Velocity under constant acceleration:

Position under constant acceleration:

Setting the standard initial conditions: \(t_i = 0\), \(x_i = x_0\), \(v_i = v_0\), and \(a(t) = a\)
Substitute into the position formula:

\[ x(t) = x_0 + v_0t + a \int_{0}^{t} t' dt' \]

\[ x(t) = x_0 + v_0t + \frac{1}{2}at^2 \]

Other useful forms (Deriving the Time-Independent Equation):

Using average velocity to find displacement:

\[ x(t) - x_0 = \frac{1}{2}(v_0 + v_f)t \]
Substitute \(t = \frac{v_f - v_0}{a}\) into the equation:

\[ x(t) - x_0 = \frac{1}{2}(v_0 + v_f) \left( \frac{v_f - v_0}{a} \right) \]

\[ x(t) - x_0 = \frac{v_f^2 - v_0^2}{2a} \]

Dimension Analysis

This shows how to find velocity when acceleration changes with time according to a power law.

Given: \(a(t) = at^\alpha\) (Note: If \(\alpha = 0 \Rightarrow\) constant acceleration)
Suppose \(\alpha > 0\).
To find velocity:

\[ v(t) = \int_{t_i}^t a(t') dt' + v_i \]

\[ = a \int_{t_i}^t (t')^\alpha dt' + v_i \]

\[ = \frac{a}{\alpha + 1} (t')^{\alpha+1} \Big|_{t_i}^t + v_i \]

\[ = \frac{a}{\alpha + 1} (t^{\alpha+1} - t_i^{\alpha+1}) + v_i \]

Rules for counting and calculating with significant figures:

Leading zeros: Not counted
Trailing zeros: Counted
Addition/Subtraction: The result is rounded to the same decimal place as the number with the first uncertain digit (least precise decimal place).
- Example:
  
  \(213.25+16.\underline{1}+ 0.124 = xxx.\underline{x}xx \rightarrow\) Result has 4 sig figs (e.g., \(229.5\))
Multiplication/Division: The result has the same number of significant figures as the term with the least sig figs.
- Example: \(325.78 \times 0.0145 \div 789.2 = 0.00599\)
  - \(325.78\) (5 sig figs)
  - \(0.0145\) (3 sig figs) \(\leftarrow\) Least sig figs
  - \(789.2\) (4 sig figs)
  - Result: \(0.00599\) (3 sig figs)

This section covers how uncertainties (errors) in individual variables propagate into the final calculated result.

For a function \(y = y(x_1, x_2, \dots, x_n)\), the absolute uncertainty \(\Delta y\) is given by:

\[\Delta y = \sqrt{\left(\frac{\partial y}{\partial x_1}\right)^2 (\Delta x_1)^2 + \left(\frac{\partial y}{\partial x_2}\right)^2 (\Delta x_2)^2 + \dots}\]

Addition (\(y = x_1 + x_2\)):

\[ \Delta y = \sqrt{(\Delta x_1)^2 + (\Delta x_2)^2} \]
Multiplication (\(y = x_1 x_2\)):

\[ \Delta y = \sqrt{(x_2 \Delta x_1)^2 + (x_1 \Delta x_2)^2} \]
- Relative uncertainty form:
  
  \[ \Rightarrow \frac{\Delta y}{|y|} = \sqrt{\left(\frac{\Delta x_1}{x_1}\right)^2 + \left(\frac{\Delta x_2}{x_2}\right)^2} \]
Division (\(y = x_1 / x_2\)):

\[ \Delta y = \sqrt{\left(\frac{1}{x_2} \Delta x_1\right)^2 + \left(\frac{x_1}{x_2^2} \Delta x_2\right)^2} \]
- Relative uncertainty form (matches multiplication):
  
  \[ \Rightarrow \frac{\Delta y}{|y|} = \sqrt{\left(\frac{\Delta x_1}{x_1}\right)^2 + \left(\frac{\Delta x_2}{x_2}\right)^2} \]

Calculating the product and uncertainty for \((5.5 \pm 0.1) \times (6.4 \pm 0.1)\):

Result: \(= 35 \pm 1\)
Uncertainty Calculation (\(\Delta y\)): The uncertainty value of \(1\) is calculated using the relative uncertainty formula rearranged for \(\Delta y\):

\[ \Delta y = \sqrt{\left(\frac{0.1}{5.5}\right)^2 + \left(\frac{0.1}{6.4}\right)^2} \times 35 \]