Motion in a Straight Line (College Board AP® Calculus AB)

Introduction to straight line motion

What is straight line motion?

• Straight line motion models how objects move in a straight line, with respect to time

  • This may be described as motion 'along the -axis'

  • The straight line will have a positive and a negative direction

    • On the -axis this will be the usual positive and negative directions

    • If the question doesn't specify, you can choose the positive and negative directions

    • (Just be consistent once you've made a choice!)

What terminology do I need to be aware of?

• Particle

  • A particle is the general term used for an object

  • A particle is assumed to be the 'size' of a single point

    • So you don't need to worry about its 3D dimensions

• Time  

  • Time is usually measured in seconds ()

  • Displacement, velocity and acceleration are all functions of time

  • 'Initial' or 'Initially' means 'when '

• Displacement 

  • is the usual notation for displacement

    • For motion along the -axis,  may be used instead of 

  • Displacement is usually measured in feet () or meters ()

    • Sometimes questions may not state a particular unit, and will simply use "units"

  • The displacement of a particle is its distance relative to a fixed point

    • The fixed point may be (but is not always) the particle’s initial position

  • Displacement will be zero, , when the object is at the fixed point

  • Otherwise the displacement will be

    • positive if the particle is in the positive direction from the fixed point

    • or negative if it is in the negative direction from the fixed point

• Distance

  • Distance is the magnitude of displacement

  • Use of the word distance could refer to

    • the distance traveled by a particle

    • the (straight line) distance the particle is from a particular point

  • Be careful not to confuse displacement with distance

    • Consider a bus starting and ending its journey at a bus depot,

    • its displacement will be zero when it returns to the depot

    • but the distance the bus has travelled will be the length of the route

  • Distance is always positive (or zero)

• Velocity 

  • Velocity is usually measured in feet or meters per second

  • The velocity of a particle is the rate of change of its displacement at time

    • Velocity will be positive if the particle is moving in the positive direction

    • Or negative if it is moving in the negative direction

  • If the particle is stationary, that means the velocity is zero, 

    • '(Instantaneously) at rest' also means that 

• Speed

  • Speed is the magnitude (i.e. absolute value or modulus) of the velocity

  • For a particle moving in a straight line

    • speed is the 'velocity ignoring the direction'

    • if,  speed 

    • if ,  speed = 

• Acceleration

  • Acceleration is usually measured in feet or meters per second squared

    • That is the same as feet or meters per second per second

  • The acceleration of a particle is the rate of change of its velocity at time

  • Acceleration can be positive or negative

    • but the sign alone cannot fully describe the particle’s motion

  • If velocity and acceleration have the same sign

    • then the particle is accelerating (speeding up)

  • if velocity and acceleration have different signs

    • then the particle is decelerating (slowing down)

  • At times when the acceleration is zero, ,

    • the particle is moving with constant velocity 

  • In all cases the direction of motion is determined by the sign (+ or -) of the velocity

  • There is no special term for the magnitude of acceleration

    • "The magnitude of the acceleration" is simply used instead

Velocity & acceleration as derivatives

What is velocity as a derivative?

• Velocity is the rate of change of displacement

  •  

  • (differentiate displacement to get velocity)

• Velocity is the slope of a displacement-time graph

What is acceleration as a derivative?

• Acceleration is the rate of change of velocity

  •  

  • (differentiate velocity to get acceleration)

• Acceleration is the slope of a velocity-time graph

• This means that acceleration is also the rate of change, of the rate of change of displacement

  • 

  • (differentiate displacement twice to get acceleration)