# Distance and Displacement

Distance and displacement are two important concepts in physics that describe the position of an object in space. While they may seem similar, they have distinct differences.

## Distance

• Distance is the total length of the path traveled by an object.

• It is a scalar quantity, meaning it has only magnitude and no direction.

• Distance is measured in units such as meters, kilometers, or miles.

• Distance is not used as much as Displacement in the AP exam, but it is denoted by the letter s or x

## Displacement

• Displacement is the change in position of an object from its initial position to its final position.

• It is a vector quantity, meaning it has both magnitude and direction.

• Displacement is measured in units such as meters, kilometers, or miles, and is represented by a vector with an arrow pointing from the initial position to the final position.

# Vector and Scalar Quantities

## Scalar Quantities

• Scalar quantities are physical quantities that have only magnitude and no direction.

• Examples of scalar quantities include mass, temperature, time, speed, distance, energy, and power.

• Scalar quantities are represented by a single number and are usually measured in units such as kilograms, seconds, meters, and joules.

## Vector Quantities

• Vector quantities are physical quantities that have both magnitude and direction.

• Examples of vector quantities include displacement, velocity, acceleration, force, and momentum.

• Vector quantities are represented by a vector, which is a quantity that has both magnitude and direction.

• Vectors are usually represented graphically as arrows, where the length of the arrow represents the magnitude of the vector and the direction of the arrow represents the direction of the vector.

• Vector quantities can be added and subtracted using vector algebra, which takes into account both the magnitude and direction of the vectors.

# Position, Velocity, and Acceleration

## Position

Position is the location of an object relative to a chosen reference point. It is a vector quantity that can be described using distance and direction. Typically, a coordinate system is used to show where an obejct is located.

• To determine which way an object is moving look at which way the Position vs Time Graph is sloped

• The slope of a Position vs Time Graph is equal to velocity

• When the slope is a straight line it has constant velocity

• When the slope is a curved lived there is a change in velocity (acceleration)

• When the slope is zero the object is at rest

• The y-intercept is the initial position of the object

# Speed vs Velocity

Speed and velocity are both terms used to describe the motion of an object, but they have different meanings.

## Speed

Speed is a scalar quantity that refers to how fast an object is moving. It is calculated by dividing the distance traveled by the time taken to travel that distance. The SI unit of speed is meters per second (m/s).

• Scalar quantity

• SI Unit: Meters (m)/Seconds (s)

Equation: S = D/t

## Velocity

Velocity is a vector quantity that refers to the rate at which an object changes its position. It is calculated by dividing the displacement of an object by the time taken to travel that displacement. The SI unit of velocity is meters per second (m/s).

• Vector quantity

• SI Unit: Meters (m)/Seconds (s)

Equation: V = x/t

A position vs time graph depicts velocity and a velocity vs time graph depicts acceleration.

# Acceleration

Acceleration is the rate of change of velocity with respect to time. It is a vector quantity, which means it has both magnitude and direction. In AP Physics 1, acceleration is an important concept that is used to describe the motion of objects.

## Calculating Acceleration

The formula for acceleration is:

``a = (v_f - v_i) / t``

where `a` is acceleration, `v_f` is final velocity, `v_i` is initial velocity, and `t` is time.

## Units of Acceleration

The SI unit of acceleration is meters per second squared (m/s^2). Other common units of acceleration include feet per second squared (ft/s^2) and kilometers per hour squared (km/h^2).

## Positive and Negative Acceleration

When an object is speeding up, its acceleration is positive. When an object is slowing down, its acceleration is negative. If an object is moving in the opposite direction of its acceleration, the acceleration is also negative.

## Uniform Acceleration

Uniform acceleration is when an object's acceleration is constant over time. This means that the object's velocity changes by the same amount in each unit of time. The formula for uniform acceleration is:

``a = (v_f - v_i) / t = (d/t) / t = d / t^2``

where `d` is the distance traveled.

## Non-Uniform Acceleration

Non-uniform acceleration is when an object's acceleration changes over time. This means that the object's velocity changes by different amounts in each unit of time. The formula for non-uniform acceleration is more complex and requires calculus.

## Free Fall

Free fall is a special case of uniform acceleration where an object is falling under the influence of gravity. The acceleration due to gravity is approximately 9.8 m/s^2 near the surface of the Earth. The formula for free fall is:

``d = (1/2)gt^2``

where `d` is the distance fallen, `g` is the acceleration due to gravity, and `t` is time.

# Uniformly Accelerated Motion and the BIG FIVE

## Uniformly Accelerated Motion

• Uniformly accelerated motion is a type of motion where the acceleration of an object remains constant.

• The velocity of the object changes at a constant rate.

• The acceleration can be positive or negative depending on the direction of the motion.

## The BIG FIVE Equations of Motion

• The BIG FIVE equations of motion are a set of equations that describe the motion of an object under uniformly accelerated motion.

• These equations relate the initial velocity, final velocity, acceleration, displacement, and time of an object.

• The equations are:

• v = u + at

• s = ut + 1/2at^2

• v^2 = u^2 + 2as

• s = 1/2(u + v)t

• a = (v - u)/t

• Here,

• u = initial velocity

• v = final velocity

• a = acceleration

• s = displacement

• t = time

## Example

• Suppose a car starts from rest and accelerates uniformly at 5 m/s^2 for 10 seconds. Find the final velocity and displacement of the car.

• Using the BIG FIVE equations of motion, we can find:

• v = u + at = 0 + 5*10 = 50 m/s

• s = ut + 1/2at^2 = 010 + 1/25*10^2 = 250 m

• Therefore, the final velocity of the car is 50 m/s and the displacement is 250 m.

# Projectile Motion and Angled Motion

## Projectile Motion

• Projectile motion is the motion of an object that is thrown or launched into the air and then moves under the influence of gravity.

• The path of a projectile is a parabolic curve.

• The horizontal and vertical components of motion are independent of each other.

• The acceleration due to gravity acts only in the vertical direction.

### Equations of Projectile Motion

• The horizontal component of velocity is constant.

• The vertical component of velocity changes due to the acceleration due to gravity.

• The time of flight is the time taken by the projectile to reach the ground.

• The maximum height reached by the projectile is given by the formula: `h = (v₀sinθ)² / 2g`

• The range of the projectile is given by the formula: `R = v₀²sin2θ / g`

## Angled Motion

• Angled motion is the motion of an object that is thrown or launched at an angle to the horizontal.

• The path of an angled projectile is a parabolic curve.

• The horizontal and vertical components of motion are dependent on each other.

• The acceleration due to gravity acts in both the horizontal and vertical directions.

### Equations of Angled Motion

• The horizontal component of velocity is given by: `v₀cosθ`

• The vertical component of velocity is given by: `v₀sinθ - gt`

• The time of flight is given by: `t = 2v₀sinθ / g`

• The maximum height reached by the projectile is given by: `h = (v₀sinθ)² / 2g`

• The range of the projectile is given by: `R = v₀²sin2θ / g`

Note: `v₀` is the initial velocity, `θ` is the angle of projection, `g` is the acceleration due to gravity, and `t` is the time taken.