Pour Le Dumbass ψ(｀∇´)ψ

1. Speed

Average Speed:

vavg=dtotalttotalv_{avg} = \frac{d_{total}}{t_{total}}

Speed:

v=dtv = \frac{d}{t}

d = distance traveled; t = time

Speed is a scalar (no direction).

2. Displacement

Displacement:

△x=xf−xi\boldsymbol{\triangle x} = \boldsymbol{x}_f - \boldsymbol{x}_i

In 1-D:

△x=xf−xi\triangle x = x_f - x_i

xi=x_i = initial position; xf=x_f = final position

Displacement includes direction.

3. Average Velocity

vavg=△x△t\boldsymbol{v}_{avg} = \frac{\boldsymbol{\triangle x}}{\boldsymbol{\triangle t}}

vavg=(xf−xi)(tf−ti)\boldsymbol{v}_{avg} = \frac{(\boldsymbol{x}_f - \boldsymbol{x}_i)}{(\boldsymbol{t}_f - \boldsymbol{t}_i)}

△t=tf−ti\boldsymbol{\triangle t} = \boldsymbol{t}_f - \boldsymbol{t}_i

In 1-D, the + or - sign gives the direction.

4. Instantaneous Velocity

v(t)=dx(t)dt\boldsymbol{v}(t) = \frac{d \boldsymbol{x}(t)}{d t}

Instantaneous velocity = slope of the tangent line on a position vs. time graph.

5. Acceleration

aavg=△v△t\boldsymbol{a}_{avg} = \frac{\boldsymbol{\triangle v}}{\boldsymbol{\triangle t}}

aavg=(vf−vi)(tf−ti)\boldsymbol{a}_{avg} = \frac{(\boldsymbol{v}_f - \boldsymbol{v}_i)}{(\boldsymbol{t}_f - \boldsymbol{t}_i)}

△t=tf−ti\boldsymbol{\triangle t} = \boldsymbol{t}_f - \boldsymbol{t}_i

a(t)=dv(t)dt\boldsymbol{a}(t) = \frac{d \boldsymbol{v}(t)}{d t}

Acceleration = slope of a velocity vs. time graph.

6. Position-Time Graph

Slope of an x vs. t graph = velocity

vavg=△x△t\boldsymbol{v}_{avg} = \frac{\boldsymbol{\triangle x}}{\boldsymbol{\triangle t}}

v=dxdt\boldsymbol{v} = \frac{d \boldsymbol{x}}{d t}

POSITION → SLOPE → VELOCITY

7. Velocity-Time Graph

Slope of a v vs. t graph = acceleration

aavg=△v△t\boldsymbol{a}_{avg} = \frac{\boldsymbol{\triangle v}}{\boldsymbol{\triangle t}}

a=dvdt\boldsymbol{a} = \frac{d \boldsymbol{v}}{d t}

Area under a velocity-time graph: △x=area under the v−t graph\boldsymbol{\triangle x} = \text{area under the } v-t \text{ graph}

VELOCITY → SLOPE → ACCELERATION

VELOCITY → AREA → DISPLACEMENT

8. Acceleration-Time Graph

Area under an acceleration-time graph: △v=area under the a−t graph\boldsymbol{\triangle v} = \text{area under the } a-t \text{ graph}

For constant acceleration: △v=a△t\boldsymbol{\triangle v} = \boldsymbol{a} \boldsymbol{\triangle t}

ACCELERATION → AREA → CHANGE IN VELOCITY

9. Constant Acceleration Equations

vf=vi+at\boldsymbol{v}_f = \boldsymbol{v}_i + \boldsymbol{a}t

△x=vit+12at2\boldsymbol{\triangle x} = \boldsymbol{v}_i t + \frac{1}{2}\boldsymbol{a}t^{2}

xf=xi+vit+12at2\boldsymbol{x}_f = \boldsymbol{x}_i + \boldsymbol{v}_i t + \frac{1}{2}\boldsymbol{a}t^{2}

vf2=vi2+2a△x\boldsymbol{v}_f^{2} = \boldsymbol{v}_i^{2} + 2\boldsymbol{a}\boldsymbol{\triangle x}

10. Average Velocity with Constant Acceleration

vavg=(vi+vf)2\boldsymbol{v}_{avg} = \frac{(\boldsymbol{v}_i + \boldsymbol{v}_f)}{2}

△x=(vi+vf)2t\boldsymbol{\triangle x} = \frac{(\boldsymbol{v}_i + \boldsymbol{v}_f)}{2}t

11. Gravity / Free Fall

g = 9.8 m/s²

If UP is positive: a=−9.8 m/s2\boldsymbol{a} = -9.8 \text{ m/s}²

If DOWN is positive: a=+9.8 m/s2\boldsymbol{a} = +9.8 \text{ m/s}²

Use the constant-acceleration equations for free-fall problems.

12. Time

△t=tf−ti\boldsymbol{\triangle t} = \boldsymbol{t}_f - \boldsymbol{t}_i

13. Direction / Sign Rules

Choose a positive direction before solving.

Right = + and Left = -

OR Up = + and Down = -

Velocity and acceleration have the SAME sign → speeding up.

Velocity and acceleration have OPPOSITE signs → slowing down.

14. Important Graph Relationships

Position vs. Time: slope = velocity

Velocity vs. Time: slope = acceleration

Velocity vs. Time: area = displacement

Acceleration vs. Time: area = change in velocity

15. Vector vs. Scalar Summary

Vectors: △x=displacement;v=velocity;a=acceleration\boldsymbol{\triangle x} = \text{displacement}; \boldsymbol{v} = \text{velocity}; \boldsymbol{a} = \text{acceleration}

Scalars: d=distance;v=speed;t=timed = \text{distance}; v = \text{speed}; t = \text{time}

MOST IMPORTANT FORMULAS TO MEMORIZE

△x=xf−xi\boldsymbol{\triangle x} = \boldsymbol{x}_f - \boldsymbol{x}_i

vavg=△x△t\boldsymbol{v}_{avg} = \frac{\boldsymbol{\triangle x}}{\boldsymbol{\triangle t}}

aavg=△v△t\boldsymbol{a}_{avg} = \frac{\boldsymbol{\triangle v}}{\boldsymbol{\triangle t}}

vf=vi+at\boldsymbol{v}_f = \boldsymbol{v}_i + \boldsymbol{a}t

△x=vit+12at2\boldsymbol{\triangle x} = \boldsymbol{v}_i t + \frac{1}{2}\boldsymbol{a}t^{2}

vf2=vi2+2a△x\boldsymbol{v}_f^{2} = \boldsymbol{v}_i^{2} + 2\boldsymbol{a}\boldsymbol{\triangle x}

vavg=(vi+vf)2\boldsymbol{v}_{avg} = \frac{(\boldsymbol{v}_i + \boldsymbol{v}_f)}{2}

△x=(vi+vf)2t\boldsymbol{\triangle x} = \frac{(\boldsymbol{v}_i + \boldsymbol{v}_f)}{2}t

g = 9.8 m/s²