你是一位物理老师，需要给学生进校这个题目的讲解，麻烦给出一个讲解视频，逐一对每个选项进行分析，给出正确的答案。（第一问轨道是竖直方向，作图时方向不要出现错误；第三问分三个阶段讨论，最后答案T=2s）---**Question 3**
**Question Stem:**
如图所示, 竖直平面内固定着光滑半圆形轨道 $AMB$, 其中直径 $AOB$ 为竖直方向, 半径 $R=1.6m$, 质量 $M=1kg$ 的长薄板静置于倾角 $\theta = 37^{\circ}$ 的粗糙斜面 $CD$ 上, 其最上端刚好在斜面顶端 $C$ 点. 一质量为 $m=1.5kg$ 的滑块 (可看作质点) 从圆轨道 $B$ 点以 $v_0 = 4\sqrt{5}m/s$ 的初速度进入轨道, 并刚好通过最高点 $A$ 点, 接着从 $A$ 点水平抛出, 恰好以平行于斜面的速度落到薄板最上端, 并在薄板上开始向下运动, 小物体落到薄板最上端时, 薄板无初速度释放并开始沿斜面向下运动, 其运动至斜面底端时, 立即与 $D$ 处挡板碰撞, 速度立即减为 0. 已知斜面 $CD$ 长 $L_2=14.75m$, 薄板长 $L_1=7.25m$, 厚度忽略不计, 其与斜面的动摩擦因数 $\mu_1=0.25$, 滑块与长薄板间的动摩擦因数 $\mu_2=0.5$. 运动过程中空气阻力均忽略不计, $g=10m/s^2$, $\sin 37^{\circ}=0.6$, $\cos 37^{\circ}=0.8$. 试求:
(1) 滑块在 $A$ 点时的初速度;
(2) $AC$ 两点间的水平距离;
(3) 滑块从 $C$ 点到达 $D$ 所需要的时间.
**Given Information:**
* Vertical plane, fixed smooth semicircle track AMB.
* Diameter AOB is vertical.
* Radius of semicircle R = 1.6m.
* Mass of thin plate M = 1kg.
* Thin plate is at rest on a rough inclined plane CD.
* Inclination angle of plane CD is θ = 37°.
* Top end of the thin plate is at point C.
* Mass of slider m = 1.5kg (can be treated as a point mass).
* Slider enters the track at point B with initial velocity v0 = 4√5 m/s.
* Slider passes through the highest point A.
* Slider is horizontally projected from A.
* Slider lands on the top end of the thin plate (point C) with velocity parallel to the inclined plane.
* When the slider lands on the plate at C, the plate is released from rest and starts moving down the incline.
* Motion stops when the plate reaches the baffle at D (velocity becomes 0).
* Length of inclined plane CD, L2 = 14.75m.
* Length of thin plate, L1 = 7.25m.
* Thickness of plate is negligible.
* Kinetic friction coefficient between the plate and the incline, μ1 = 0.25.
* Kinetic friction coefficient between the slider and the thin plate, μ2 = 0.5.
* Air resistance is negligible.
* g = 10m/s².
* sin 37° = 0.6.
* cos 37° = 0.8.
**Sub-questions:**
(1) The initial velocity of the slider at point A.
(2) The horizontal distance between points A and C.
(3) The time required for the slider to travel from point C to point D.
**Chart/Diagram Description:**
* **Type:** Combined schematic diagram showing a circular track connected to an inclined plane with a block on it.
* **Main Elements:**
* A semicircle AMB is shown on the left in a vertical plane.
* Points A, O, B, M are labeled on or near the semicircle.
* AOB is a vertical line segment representing the diameter, with A at the top and B at the bottom. O is the center.
* A vector labeled v0 is shown originating from B, pointing horizontally to the right. A right angle symbol is shown between v0 and the vertical line AOB at B.
* An inclined plane CD is shown on the right, starting from point C and extending downwards to point D.
* A line segment connecting A and C is shown as a dashed curved line, representing the trajectory of the slider after leaving A.
* A thin rectangular block (representing the thin plate) is placed on the inclined plane, with its top end at C. The length of the block is shown as extending downwards along the incline.
* The angle of inclination θ is shown at point D, between the inclined plane CD and the horizontal line.
* **Relative Position and Direction:**
* The semicircle is to the left of the inclined plane.
* Point A is above point O and B. Point B is below O and A. AOB is vertical.
* Point C is at the top of the inclined plane.
* The thin plate is on the inclined plane, starting at C.
* The dashed line from A to C indicates projectile motion.
* The velocity vector v0 at B is horizontal.
* The inclined plane slopes downwards from left to right.
* **Labels and Annotations:** Points A, B, M, O, C, D, v0, θ are labeled. R, L1, L2 are labeled (though not explicitly shown in the diagram, they are mentioned in the text and relate to the diagram). A right angle symbol is shown at B. A dashed curved line from A to C is labeled.