Essential Equations and Concepts for Kinematics and Forces

Motion Quantities

Displacement, Velocity, and Acceleration

Understanding motion in physics begins with the definitions of displacement, velocity, and acceleration. These vector quantities describe how an object's position changes over time.

• Displacement (\( \Delta \vec{r} \)): The change in position of an object.

• Velocity (\( \vec{v} \)): The rate of change of displacement with respect to time.

• Acceleration (\( \vec{a} \)): The rate of change of velocity with respect to time.

Key Equations:

• Displacement: $\Delta \vec{r} = \vec{r}_2 - \vec{r}_1$

• Velocity: $\vec{v} = \frac{d\vec{r}}{dt}$

• Acceleration: $\vec{a} = \frac{d\vec{v}}{dt}$

Example: If a car moves from position \( \vec{r}_1 = (2, 3) \) m to \( \vec{r}_2 = (5, 7) \) m, its displacement is \( (3, 4) \) m.

1D Motion

General Equations for One-Dimensional Motion

One-dimensional motion describes objects moving along a straight line, often with varying acceleration.

• Velocity as a function of time: $v_x = v_{x0} + \int_{0}^{t} a_x(t) dt$

• Position as a function of time: $x = x_0 + \int_{0}^{t} v_x(t) dt$

Example: If acceleration is constant, these integrals simplify to the constant acceleration equations below.

Constant Acceleration

Kinematic Equations for Constant Acceleration

When acceleration is constant, the equations of motion become especially useful for solving problems in kinematics.

• $x = x_0 + v_{0x} t + \frac{1}{2} a_x t^2$

• $v_x = v_{0x} + a_x t$

• $v_x^2 = v_{0x}^2 + 2 a_x \Delta x$

Example: A ball thrown upward with initial velocity \( v_{0x} = 10 \) m/s and acceleration \( a_x = -9.8 \) m/s2 will reach its maximum height when \( v_x = 0 \).

Circular Motion

Radial and Tangential Acceleration

Circular motion involves objects moving along a curved path, requiring special consideration of acceleration components.

• Radial (centripetal) acceleration: $a_r = \frac{v^2}{r}$

• Tangential acceleration: $a_t = \frac{dv}{dt}$

Example: A car turning in a circle of radius 20 m at 10 m/s has a radial acceleration of $a_r = \frac{(10)^2}{20} = 5$ m/s2.

Forces

Newton's Second Law and Weight

Forces cause changes in motion, described by Newton's laws. The force of gravity is a common example.

• Newton's Second Law: $\sum \vec{F}_i = m \vec{a}$

• Weight (force of gravity): $\vec{F}_g = m \vec{g}$

• Standard gravitational acceleration: $g_E = 9.8 \ \mathrm{m/s^2}$

Example: The weight of a 2 kg object on Earth is $F_g = 2 \times 9.8 = 19.6$ N downward.

Trigonometry

Basic Trigonometric Relationships

Trigonometry is essential for resolving vectors and analyzing forces at angles.

• $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

• $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

• $\sin \theta = \cos(90^\circ - \theta)$

• $\sin 2\theta = 2 \sin \theta \cos \theta$

Example: For a right triangle with sides 3, 4, and 5, $\sin \theta = 3/5$ if the opposite side is 3 and the hypotenuse is 5.

Quadratic Equations

Solving Quadratic Equations in Physics

Quadratic equations frequently arise in kinematics, especially when solving for time or displacement.

• General form: $a x^2 + b x + c = 0$

• Solution: $x = \frac{ -b \pm \sqrt{b^2 - 4ac} }{2a}$

Example: To find the time when a projectile returns to the ground, set $y = 0$ and solve the quadratic equation for $t$.