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在透镜成象中,设物、象沿主轴移动的速度分别为V_物=du/dt,V_象=dv/dt,并将高斯公式:1/u+1/v=1/f对时间求导数,则 (-du/dt)/u~2+(-dv/dt)/v~2=0,即V_象/V_物=-(v/u)~2。那么,怎样引导学生不用求导而通过v—u图象的物理意义得出相同结果呢?下面以凸透镜为例,从速度的方向、大小和参照物分叙如下。 1 由高斯公式,v—u图象是关于点(f,f)对称的双曲线,v=f,u=f分别是它的水平渐近线和垂直渐近线。在曲线上任取P_1(u_1,v_1),P_2(u_2,v_2)两点,过P_1P_2的直线斜率为:
In the lens imaging, the velocity of the object and the image moving along the spindle is respectively V_object=du/dt, V_image=dv/dt, and the Gaussian formula: 1/u+1/v=1/f The time derivative is (-du/dt)/u~2+(-dv/dt)/v~2=0, that is, V_image/V_object=-(v/u)~2. Then, how can we guide students to obtain the same results from the physical meaning of v-u images without guiding? The following uses the convex lens as an example to classify the velocity direction, size, and reference object as follows. 1 From the Gaussian formula, the v-u image is a hyperbolic symmetric about the point (f, f), v = f, and u = f are its horizontal asymptote and vertical asymptote, respectively. P_1(u_1,v_1), P_2(u_2,v_2) are two points on the curve, and the slope of the line passing through P_1P_2 is: