Wednesday, October 11, 2006
Uniformly-Accelerated Motion and The Big Five (p. 17 PR)
The simplest type of motion to analyze is motion in which the acceleration is constant. Although true uniform acceleration is rarely achieved in the real word, many common motions are governed by approximately constant acceleration and, in these cases, the kinematics of uniformly accelerated motion provide a pretty good description of what's happening. Notice that if the acceleration is constant, then taking an average yields nothing new, so a = a.
Another restriction that will make our analysis easier is to consider only motion that takes place along a straight line. In these cases, there are only two possibly directions of motion. One is positive and the opposite direction is negative. Most of the quantities we've been dealing with--displacement, velocity, and acceleration--are vectors, which means that they include both a magnitude and a direction. With straight-line motion, direction can be specificied simply by adding a + or - sign to the magnitude of the quantity.
The fundamental quantities are displacement (Δs), velocity (v), and acceleration (a). Acceleration in a change in velocity, from an initial velocity (v0 or vi) to a final velocity (vf or simply v--with no subscript). And, finally, the motion takes place during some elapsed time interval, Δt. Therefore, we have five kinematic quantities: Δs, v0, v, a, and Δt.
These five quantities are related by a group of five equations that we call The Big Five. They work in cases where acceleration is uniform, which are the cases we're considering.
Big Five #1
Δs = vΔt
Variable that's missing: a
Big Five #2
Δv = aΔt
Variable that's missing: Δs
Big Five #3
Δs = v0Δt + 0.5a(Δt)2
Variable that's missing: v
Big Five #4
Δs = vΔt - 0.5a(Δt)2
Variable that's missing: v0
Big Five #5
v2 = v02 + 2aΔs
Variable that's missing: Δt
Another restriction that will make our analysis easier is to consider only motion that takes place along a straight line. In these cases, there are only two possibly directions of motion. One is positive and the opposite direction is negative. Most of the quantities we've been dealing with--displacement, velocity, and acceleration--are vectors, which means that they include both a magnitude and a direction. With straight-line motion, direction can be specificied simply by adding a + or - sign to the magnitude of the quantity.
The fundamental quantities are displacement (Δs), velocity (v), and acceleration (a). Acceleration in a change in velocity, from an initial velocity (v0 or vi) to a final velocity (vf or simply v--with no subscript). And, finally, the motion takes place during some elapsed time interval, Δt. Therefore, we have five kinematic quantities: Δs, v0, v, a, and Δt.
These five quantities are related by a group of five equations that we call The Big Five. They work in cases where acceleration is uniform, which are the cases we're considering.
Big Five #1
Δs = vΔt
Variable that's missing: a
Big Five #2
Δv = aΔt
Variable that's missing: Δs
Big Five #3
Δs = v0Δt + 0.5a(Δt)2
Variable that's missing: v
Big Five #4
Δs = vΔt - 0.5a(Δt)2
Variable that's missing: v0
Big Five #5
v2 = v02 + 2aΔs
Variable that's missing: Δt
// posted by H. Brown @ 9:39 PM 