One-dimensional kinematics (the description of 1D motion)
Vector kinematics with applications
THEORY
There are six kinematic terms. These definitions (really all definitions) should be memorized.
● time, t … reading of a stopwatch
● time interval, (t ( tf - ti … difference in two times
● position, r … vector from an origin to the location of a particle
r(t) = x(t) i + y(t) j
graph of r
● displacement, (r ( rf - ri … difference in two positions
graph of (r
● velocity, v ( (r/(t … this is average velocity over (t,
in the limit (t( 0, this becomes
v ( dr/dt = dx/dt i + dy/dt j = vxi + vyj
v points parallel to the tangent to the path of motion
● acceleration, a ( (v/(t … this is average acceleration over (t,
in the limit (t( 0, this becomes
a ( dv/dt = d2r/dt2 = d2x/dt2 i + d2y/dt2 j = dvx/dt i + dvy/dt j
About speed
( speed is the magnitude of the velocity, ( speed ( 0
APPLICATIONS OF VECTOR KINEMATICS
( projectile motion
Place the origin of coordinates with x-axis horizontal and y-axis vertical (up). In PM there is zero horizontal acceleration and constant (-g) vertical acceleration. The key is each direction behaves independently. These equations are analogous to the free fall equations in one dimension.
| x- direction y-direction |
| |
|ax = 0 ay = -g |
| |
|vx(t) = vx(0) vy(t) = vy(0) – gt |
| |
|x(t) = x(0) + vx(0)t y(t) = y(0) + vy(0)t – ½ gt2 |
| |
|vx(0) = v0cosθ vy(0) = v0sinθ |
Applications of the above equations can be used to obtain:
the path of the projectile, i.e. the trajectory, y(x), [eliminate t from x(t) and y(t)]
the maximum elevation or height [ use vy(tmax) = 0 ]
and the maximum horizontal distance, the range. [use y=y0 and x-x0 = R ]
( uniform circular motion
We will prove in class that for motion in a circle of radius, r, with constant speed, v, the acceleration vector is
a = (v2 / r) [-r(hat)] , this latter vector points to the
center of the circle ( a is centripetal (center seeking) acceleration.
Proof
Start with, r(t) = rcosωt i + rsinωt j where ω = v/r = 2π/T
( relative velocity
We will show vPΣ = vPΣ’ + vΣ’Σ
This implies Galilean invariance of acceleration.
Proof
EXAMPLES [in class]
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