Acceleration measures how quickly velocity changes over time, considering both speed and direction. It's crucial for understanding motion in physics, from rockets launching to cars braking. Positive and negative acceleration can speed up or slow down objects in different directions.

Calculating acceleration involves comparing initial and final velocities over a time interval. The signs of velocity and acceleration are important, indicating whether an object is speeding up or slowing down. This connects to forces and Newton's Second Law, which explains the relationship between force, mass, and acceleration.

Acceleration and Velocity

Acceleration and velocity changes

• Acceleration measures the rate at which velocity changes over time
  • Velocity includes both speed (magnitude) and direction (vector quantity)
  • Changes in velocity result from changes in speed, direction, or both (train accelerating, car turning)
• Positive acceleration occurs when an object speeds up in the positive direction or slows down in the negative direction (rocket launch, applying brakes while reversing)
• Negative acceleration occurs when an object slows down in the positive direction or speeds up in the negative direction (parachute deployment, pressing gas pedal while reversing)
• Acceleration can change the direction of motion without changing speed
  • Uniform circular motion maintains constant speed but continuously changes direction (satellite orbiting Earth, rider on a merry-go-round)
• Displacement is the change in position of an object, which is directly related to velocity and acceleration

Calculation of acceleration

• Average acceleration formula: $a = \frac{v_f - v_i}{t}$
  • $a$ represents average acceleration
  • $v_f$ represents final velocity
  • $v_i$ represents initial velocity
  • $t$ represents time interval
• Calculate acceleration by subtracting initial velocity from final velocity and dividing by time interval (car accelerating from 20 m/s to 30 m/s in 5 seconds, $a = \frac{30 m/s - 20 m/s}{5 s} = 2 m/s^2$)
• Acceleration units are typically meters per second squared ($m/s^2$)
• Instantaneous acceleration is found by taking the limit of average acceleration as time interval approaches zero (calculus concept)

Signs of velocity and acceleration

• Velocity sign indicates direction of motion
  • Positive velocity signifies movement in the positive direction (right or up)
  • Negative velocity signifies movement in the negative direction (left or down)
• Acceleration sign indicates whether an object is speeding up or slowing down
  • Positive acceleration
    • Object speeds up when velocity is positive (pressing gas pedal while driving forward)
    • Object slows down when velocity is negative (releasing gas pedal while reversing)
  • Negative acceleration
    • Object slows down when velocity is positive (applying brakes while driving forward)
    • Object speeds up when velocity is negative (pressing gas pedal while reversing)
• Object speeds up when velocity and acceleration have the same sign (both positive or both negative)
• Object slows down when velocity and acceleration have opposite signs (one positive, one negative)

Forces and Motion

• Force is the cause of acceleration, as described by Newton's Second Law
• Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass
• Mass is a measure of an object's resistance to acceleration
• Kinematics describes the motion of objects using concepts like displacement, velocity, and acceleration without considering the causes of motion