AP physics 1

1. Key Quantities

• Position ($x$ or $y$): Location of an object relative to a reference frame, measured in meters ($\text{m}$).

• Displacement ($\Delta x$): Vector change in position ($\Delta x = x<em>f - x</em>i$$).

• Distance ($d$): Scalar total path length traveled.

• Velocity ($v$): Rate of change of position (vector).

  • Average Velocity: $v_{\text{avg}} = \frac{\Delta x}{\Delta t}$

  • Instantaneous Velocity: Velocity at a single point in time.

• Speed: Magnitude of velocity (scalar).

• Acceleration ($a$): Rate of change of velocity (vector), measured in $m/s^2$.

  • Average Acceleration: $a<em>{\text{avg}} = \frac{\Delta v}{\Delta t} = \frac{v</em>f - v_i}{\Delta t}$

2. Kinematic Equations (Constant Acceleration)

1. Velocity-Time Equation

   • $v = v_0 + a t$

2. Position-Time Equation

   • $x = x<em>0 + v</em>0 t + \frac{1}{2} a t^2$

3. Velocity-Displacement Equation

   • $v^2 = v<em>0^2 + 2 a (x - x</em>0)$

4. Average Velocity Equation

   • $\Delta x = \left(\frac{v_0 + v}{2}\right) t$

3. Graphical Analysis

• Position vs. Time ($x$ vs. $t$):

  • Slope: Velocity ($v$).

  • Concavity: Curve bending upward indicates positive acceleration ($a > 0$); curve bending downward indicates negative acceleration ($a < 0$).

• Velocity vs. Time ($v$ vs. $t$):

  • Slope: Acceleration ($a$).

  • Area Under Curve: Displacement ($\Delta x$).

• Acceleration vs. Time ($a$ vs. $t$):

  • Area Under Curve: Change in velocity ($\Delta v$).

4. Free Fall

• Object moves under the influence of gravity alone ($a_y = -g = -9.8\,m/s^2$$).

• At peak height for an upward launch, instantaneous velocity $v = 0\,m/s$, but acceleration remains $a = -9.8\,m/s^2$.

• Symmetric trajectory: time to reach top equals time to fall back to launch height.

5. Projectile Motion

• Two-dimensional motion with independent horizontal and vertical components.

• Horizontal Motion ($x$): Zero acceleration ($a_x = 0\,m/s^2$), constant velocity.

  • $v<em>x = v</em>{0x} = v_0 \cos(\theta)$

  • $\Delta x = v_{0x} t$

• Vertical Motion ($y$): Constant acceleration ($a_y = -g = -9.8\,m/s^2$).

  • $v<em>{0y} = v</em>0 \sin(\theta)$

  • $v<em>y = v</em>{0y} - g t$

  • $\Delta y = v_{0y} t - \frac{1}{2} g t^2$

• Time ($t$) links both horizontal and vertical motion equations.