2.第二类问题已知加速度和初始条件,求υ,F A particle moves along the direction ofx axis, a=2t. At t=0, xo=0, vo=0. What are its velocity and position at t=2s? Solution: Acceleration. a=2t is not constant dy= tdt dv= tdt 0 0 dt dx=tdti dx=lidt ; x 0 3 s, 8/=267m 3 Eliminate t from1)and(2), we can also get: v=v(x);=(3x) 2/3
2. 第二类问题 已知加速度和初始条件，求 , r v dv = 2tdt A particle moves along the direction of x axis, a=2t. At t =0, x0=0，v0=0. What are its velocity and position at t =2s? Solution: Acceleration, a=2t, is not constant. (1) d d d 2 d ; 2 0 0 t t x v t t v v t = = = = = x t x t t x t t 0 2 0 2 d d ; d d (2) 3 1 ; 3 x = t Eliminate t from (1) and (2), we can also get: 2/ 3 v = v(x) ; v = (3x) 2.6 7m 3 8 So, v = 4m/s; x = =
Example: A particle moves along x direction, a=2+6x Att=0, xo=0, vo=0. What is its v(x)? dy Solution: a= dt s dv=(2+6x)dt dvdνdxdv ad= vav dt dx dt dκ (2+6x)dx= vdv 2(x+x) 0 2 ν=2yx+x << 99 no physical meaning Note: If a(v) is given, we may also find v(x) by dy dx dy dy vdy av dtdt dx dx
v x t t v a ; d (2 6 )d d d 2 = = + adx = vdv t v t v a d d d d = = x v v d d = dx dx Solution: + = x v x x v v 0 0 2 (2 6 )d d 3 2 2 1 2(x +x ) = v 3 v = 2 x + x “-” no physical meaning If a(v) is given, we may also find v(x) by x v t x t v a v d d d d d d ( ) = = ; d d x v = v Note: A particle moves along x direction, . At t =0, x0=0，v0=0. What is its v(x)? 2 a = 2 + 6x ( ) d d a v v v x = Example:
Chapter 4&5: Dynamics Newton's laws Page77-132
Chapter 4&5: Dynamics: Newton’s Laws Page77-132
Newton's Second law dp d(mv) dm d y+m dt dt dt Where p=m; is momentum(动量) of bod So in classical mechanics, it can be expressed as: net=a n rectangular coordinate: net.x net,y net
Newton’s Second Law Fnet ma = t P F d d = = = t mv d d( ) v t m d d t v m d d + P mv Where is momentum ( = 动量) of body. So in classical mechanics, it can be expressed as: net z z net y y net x x F ma F ma F ma = = = , , , In rectangular coordinate:
Applying Newton's Laws Problem Solving guide: (p94-95: problem solving 1. Pick a body based on the feature of problem(研究对 象); 2. Draw a free-body diagram(画出隔离体的示力图p89); 3. Choose coordinate axes(建立较方便的坐标系); 4. Resolve vectors along chosen axes(沿轴分解矢量) 5. Apply newton’ s laws of motion(列方程,统一单位,求 解); 6. Decide whether or not the solution makes sense
Applying Newton’s Laws 1. Pick a body based on the feature of problem (研究对 象); 2. Draw a free-body diagram (画出隔离体的示力图p89); 3. Choose coordinate axes (建立较方便的坐标系); 4. Resolve vectors along chosen axes (沿轴分解矢量); 5. Apply Newton’s laws of motion (列方程,统一单位,求 解); 6. Decide whether or not the solution makes sense. Problem Solving Guide:(p94 –95: problem solving)