Chapter 2: Motion along a Straight Line

  • Average velocity during a particular time interval for a particle is a vector quantity whose x-component is the change in x divided by the time interval.
  • The x-component of average velocity, or average x-velocity, is the x component of displacement,Δx, divided by the time interval,Δt, during which the displacement occurs.
  • The rule for the sign of velocity depends on four main case scenarios:

   

  • For example, let’s say that a particle is at point 19 m from the origin after 1s of moving, and 4.0 s after the start it is at point 277 m from the origin.
  • The x-component of the displacement is the change in the value of x (277 m - 19 m) = 258 m, that took place during the time interval of (4s-1s)=3s.
  • We define the particle’s average velocity during this time interval as: (258m)/(3s)=86 m/s.
    • The velocity at a specific instant of time or specific point along the path is known as instantaneous velocity.
    • ==In the language of calculus, the limit of Δx/Δt as Δt approaches zero is called the derivative of x with respect to t and is written dx/dt.==
  • The instantaneous velocity is the limit of the average velocity as the time interval approaches zero.
  • It equals the instantaneous rate of change of position with time.

   

  • On a graph of position as a function of time for straight line motion, the instantaneous x-velocity at any point is equal to the slope of the tangent to the curve at that point.
    • If the tangent to the x-t curve slopes upward to the right, then its slope is positive, the x-velocity is positive, and the motion is in the positive x-direction.
    • If the tangent slopes downward to the right, the slope of the x-t graph and the x-velocity are negative, and the motion is in the negative x-direction.
    • When the tangent is horizontal, the slope and the x-velocity are zero.
  • A motion diagram shows the particle’s position at various instants as well as arrows to represent the particle’s velocity at each instant.
  • The change in the x-component of velocity Δv(x), divided by the time interval Δt is known as average acceleration.

   

  • The instantaneous acceleration is the limit of the average acceleration as the time interval approaches zero. In the language of calculus, instantaneous acceleration equals the derivative of velocity with time.

 

  • The rules for the sign of the acceleration are:

   

  • On a graph of x-velocity as a function of time, ==the instantaneous x-acceleration at any point is equal to the slope of the tangent to the curve at that point==.
    • Tangents drawn at different points along the curve have different slopes, so the instantaneous x-acceleration varies with time.
    • The second derivative of any function is directly related to the concavity or curvature of the graph of that function.
    • Here, the acceleration is the double derivative of dx/dt.

     

  • Where the x-t graph is concave up (curved upward), the x-acceleration is positive and is increasing.
  • At a point where the x-t graph is concave down (curved downward), the x-acceleration is negative and is decreasing.
  • At a point where the x-t graph has no curvature, such as an inflection point, the x-acceleration is zero and the velocity is not changing.
    • The three equations for motion in a straight line and in constant acceleration are:

   

  Here the vx is the final velocity, v0x is the initial velocity, ax is the acceleration, t is the time, x is the final position and x0 is the starting position.

  • Free fall is a case of motion with constant acceleration.
    • The magnitude of the acceleration due to gravity is a positive quantity, g.
    • The acceleration of a body in free fall is always downward.
    • ==Here, g=9.8 m/s2.==
  • When the acceleration is variable(not constant), ==we have to integrate the acceleration at each point in the distance==. Thus the equations become: