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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