Study Notes on Motion in a Straight Line

Stopping Distance ( $s$ ): The distance a vehicle travels before stopping when brakes are applied. It is calculated using $v^2 = u^2 + 2as$ . With final velocity $v = 0$ and deceleration $-a$ , the formula is $s = \frac{u^2}{2a}$ . Clearly, if speed is doubled, the stopping distance becomes four times greater.
Bullet and Planks Application: - If a bullet loses $(1/n)$ of its velocity passing through one plank, the minimum number of planks required to stop it is $m = \frac{n^2}{2n - 1}$ . - If the velocity becomes $(1/n)$ of the initial velocity after one plank, the number of planks required to stop it is $m = \frac{n^2}{n^2 - 1}$ . - If a bullet loses $(1/n)$ of its velocity over distance $x$ , the further distance $y$ traveled before rest is $y = \frac{x(n-1)^2}{2n - 1}$ .

Acceleration as a function of time $a = f(t)$ : - $dv = f(t)dt$ - $v = u + \textstyle \int f(t)dt$
Acceleration as a function of distance $a = f(x)$ : - $v \frac{dv}{dx} = f(x)$ - $v^2 = u^2 + 2 \textstyle \int f(x)dx$

Definition: The uniform acceleration of a freely falling body toward the center of the earth.
Key Characteristics: It is independent of the size, shape, or material of the body.
Values: - Maximum at the poles: $\approx 9.83\,m/s^2$ . - Minimum at the equator: $\approx 9.78\,m/s^2$ . - Moon's gravity: $g_{\text{moon}} = \frac{1}{6} g_{\text{earth}}$ . - Center of the earth: $g = 0$ .
Sign Convention: $g$ is positive when falling toward earth (velocity increases) and negative when projected upward (velocity decreases).

Vertically Downward: - $v = u + gt$ - $h = ut + \frac{1}{2}gt^2$ - $v^2 - u^2 = 2gh$
Free Fall ( $u = 0$ ): - Final velocity at ground: $v_f = \sqrt{2gH}$ . - Time of descent: $T = \sqrt{\frac{2H}{g}}$ . - Ratio of distances in successive seconds ( $1^{st}, 2^{nd}, 3^{rd}...$ ): $1:3:5:7...$ - Ratio of total distances in intervals ( $1\,s, 2\,s, 3\,s...$ ): $1:4:9:16...$

Dynamics: The motion is decelerated during ascent ( $a = -g$ ) and accelerated during descent ( $a = +g$ ).
Maximum Height ( $H_{\max}$ ): $H_{\max} = \frac{u^2}{2g}$ .
Time of Flight ( $T$ ): $T = \frac{2u}{g}$ . Time of ascent ( $t_a$ ) equals time of descent ( $t_d$ ), where $t_a = t_d = \frac{u}{g}$ .
Velocity at Heights: - Velocity at half maximum height: $v = \frac{u}{\sqrt{2}}$ . - Velocity at $\frac{3}{4}$ of maximum height: $v = \frac{u}{2}$ .

Acceleration Changes: - Upward journey retardation: $(g + a)$ . - Downward journey acceleration: $(g - a)$ .
Comparison: - Time of descent ( $t_2$ ) is greater than time of ascent ( $t_1$ ): t_2 > t_1. - The speed of the body when it reaches the point of projection is less than the initial speed of projection.

Splash in a Well: Total time to hear the splash is $t = \sqrt{\frac{2h}{g}} + \frac{h}{v_{\text{sound}}}$ .
Elevator Motion: Time of flight inside an elevator accelerating upward is $t = \frac{2u}{g + a}$ ; if accelerating downward, $t = \frac{2u}{g - a}$ .
Three Bodies from Tower: If three bodies are projected from height $h$ (one up with $u$ , one down with $u$ , one dropped), the tower height is related to their ground-reaching times ( $t_1, t_2, t_3$ ) by $h = \frac{1}{2}gt_1 t_2$ and $t_3 = \sqrt{t_1 t_2}$ .
Frictionless Wire in Sphere: The time for a ball to slip along a wire chord inside a sphere of radius $R$ is $t = 2\sqrt{\frac{R}{g}}$ .

Differentiation: - $v = \frac{ds}{dt}$ - $a = \frac{dv}{dt} = v\frac{dv}{ds}$
Integration: - $s = \textstyle \int v\,dt$ - $\Delta v = \textstyle \int a\,dt$
Conversion Chain: - $s-t \rightarrow v-t \rightarrow a-t$ (via Differentiation). - $a-t \rightarrow v-t \rightarrow s-t$ (via Integration).