Kinematics 88-2 Theorem of composition of velocities of a particle The relationship among the absolute, relative and convected velocity will be established and stated in the theorem of composition of velocities of a particle Proof When t t+△tABA'B MM Ⅱ(t+△ Equivalent to M M1 M I(t) M is the absolute path M is the absolute displacement MM is the relative path MIM is the relative displacement MMEMM+M.M Divided the above equation by At When A→>0 gives lim4M=l0arA0△t +lim △→0△t △t→0
21 §8-2 Theorem of composition of velocities of a particle The relationship among the absolute, relative and convected velocity will be established and stated in the theorem of composition of velocities of a particle. ＝ MM1 MM ' ＋ ' M1 M When t t+△t AB A'B' M M' Equivalent to M M１ M´ MM ' is the absolute path MM ' is the absolute displacement M1M ' is the relative path M1M ' is the relative displacement t M M t MM t MM t t t = + → → → 1 0 1 0 0 lim lim lim Divided the above equation by , t When t →0 ,gives 1. Proof
运动学 §8-2点的速度合成定理 速度合成定理将建立动点的绝对速度,相对速度和牵连速度 之间的关系。 当tt△tABA'B′ 证明 MM T(t+△t 也可看成MM1M MM′为绝对轨迹 I(t) MM为绝对位移 MM为相对轨迹 M1M!为相对位移 MM=M+MM 将上式两边同除以Mt后, 取→0时的极限,得imMM"=lmMM+1mM1M △→0AtA→>0t △t→0
22 §８－２点的速度合成定理 速度合成定理将建立动点的绝对速度，相对速度和牵连速度 之间的关系。 ＝ MM1 MM ' ＋ ' M1 M 当t t+△t AB A'B' M M' 也可看成M M１ M´ MM ' 为绝对轨迹 MM ' 为绝对位移 M1M ' 为相对轨迹 M1M ' 为相对位移 t M M t MM t MM t t t = + → → → 1 0 1 0 0 lim lim lim 将上式两边同除以 t 后， 取 t →0 时的极限，得 一．证明
Kinematics I(t+1) Mi
23
运动学 I(t+1) Mi 24
24
Kinematics ∴n=V+ Conclusion: At any instant of time, the absolute velocity of a moving point equals to the geometric sum of its relative velocity and convected velocity This is the theorem of composition of velocities of a particle. Explanations: va absolute velocity of the moving point; v--relative velocity of the moving point veconvected velocity of the moving point which is the velocity of a point (convected point)on the McS I)When MCs is in translation, all points in it have the same velocity IDWhen MCs rotates, ve must be the velocity of the point in the mCs coinciding with the moving point 25
25 Explanations： va—absolute velocity of the moving point; vr—relative velocity of the moving point; ve—convected velocity of the moving point, which is the velocity of a point (convected point) on the MCS; I) When MCS is in translation, all points in it have the same velocity. II) When MCS rotates, ve must be the velocity of the point in the MCS coinciding with the moving point. Conclusion: At any instant of time, the absolute velocity of a moving point equals to the geometric sum of its relative velocity and convected velocity. This is the theorem of composition of velocities of a particle. a e r v =v +v