Acceleration (College Board AP® Physics 1: Algebra-Based): Revision Note

Average acceleration

Acceleration

• Acceleration is defined as:

The rate of change in an object's velocity

• Therefore, acceleration describes how an object's velocity changes over time

• Any change in velocity is an acceleration

• Velocity is a vector quantity with both magnitude and direction

• Therefore, an acceleration can be:

  • a change in the magnitude of an object's velocity (its speed)

  • a change in direction

• Acceleration itself is a vector quantity with both magnitude and direction

Average and instantaneous acceleration

• Instantaneous acceleration is the acceleration of an object at a specific instant of time

• Average acceleration describes a whole period of acceleration

  • For example, a train travels at a steady speed for most of its journey, but it accelerates as it leaves a station

• Average acceleration considers the initial and final states of an object over an interval of time

• In other words, the change in velocity over the time interval for which the acceleration occurred

${\overrightarrow{a}}_{avg}=\frac{∆\overrightarrow{v}}{∆t}$

• Where:

  • ${\overrightarrow{a}}_{avg}$ = average acceleration, measured in $\mathrm{m}/{\mathrm{s}}^2$

  • $∆\overrightarrow{v}$ = change in velocity, measured in $\mathrm{m}/{\mathrm{s}}^2$

  • $∆t$ = time interval, measured in $\mathrm{s}$

• If the train had an initial velocity of zero at the station, a final velocity of 50 meters per second in the positive direction, and the period of that acceleration was 60 seconds, then its average acceleration would be 0.83 meters per second squared

${\overrightarrow{a}}_{avg}=\frac{50}{60}=0.83\mathrm{m}/{\mathrm{s}}^2$

• Calculating the average acceleration over a very small time interval yields a value that is very close to the instantaneous acceleration

Negative acceleration

• Since acceleration is a vector quantity, it can have a positive or negative value

• The negative or positive value of acceleration does not always describe whether the object is speeding up or slowing down

  • The $\pm$ sign of a position value describes where the object is

  • The $\pm$ sign of a velocity value describes the direction in which the object is moving

  • The $\pm$ sign of an acceleration value only consistently describes the direction of the acceleration vector

• When a ball is thrown vertically into the air, the only force acting on the ball is weight, the product of its mass and the acceleration due to gravity at Earth's surface

• Neither the mass of the ball nor the acceleration due to gravity change during the ball's journey

• The velocity, however, does change both in magnitude and direction

  • The direction of the initial motion is in an upward, positive direction

  • As the ball gains height, it loses speed, so the magnitude of its velocity decreases

  • At the instant that the ball changes direction, its speed is zero

  • As the ball falls, the direction of motion is in a downward, negative direction

  • As the ball loses height, it gains speed, so the magnitude of its velocity increases

• The direction of the acceleration is constant in the downward, negative direction

• The magnitude of the acceleration is constant (if air resistance is ignored)

• This example highlights a case in which the $\pm$ sign of the acceleration value does not describe the speeding up and slowing down of the object