根据上述题目和解答制作视频---**Extraction Content:**
**Question Number:** 12.
**Question Stem:**
如图, 一长度 l=1.0m 的均匀薄板初始时静止在一光滑平台上, 薄板的右端与平台的边缘 O 对齐。薄板上的一个小物块从薄板的左端以某一初速度向右滑动, 当薄板运动的距离 Δl = l/6 时, 物块从薄板右端水平飞出; 当物块落到地面时, 薄板中心恰好运动到 O 点。已知物块与薄板的质量相等。它们之间的动摩擦因数 μ=0.3, 重力加速度大小 g=10m/s². 求
(1) 物块初速度大小及其在薄板上运动的时间;
(2) 平台距地面的高度。
**Translation of Question Stem:**
As shown in the figure, a uniform thin board of length l = 1.0m is initially at rest on a smooth platform, with the right end of the board aligned with the edge O of the platform. A small block on the board slides to the right from the left end of the board with a certain initial velocity. When the board has moved a distance Δl = l/6, the block flies horizontally from the right end of the board; when the block lands on the ground, the center of the board has just moved to point O. It is known that the mass of the block is equal to the mass of the board. The coefficient of kinetic friction between them is μ = 0.3, and the magnitude of acceleration due to gravity is g = 10m/s². Find:
(1) The magnitude of the initial velocity of the block and the time it moves on the board;
(2) The height of the platform from the ground.
**Questions:**
(1) 物块初速度大小及其在薄板上运动的时间 (Magnitude of the initial velocity of the block and the time it moves on the board);
(2) 平台距地面的高度 (Height of the platform from the ground).
**Given Information:**
* Thin board length: l = 1.0 m
* Initial state: Board at rest, right end aligned with platform edge O.
* Block starts from the left end of the board with an initial velocity to the right.
* When board moves distance Δl = l/6, block leaves the right end horizontally.
* When block lands on ground, board center is at O.
* Mass of block = Mass of board.
* Coefficient of kinetic friction between block and board: μ = 0.3
* Acceleration due to gravity: g = 10 m/s²
* Platform is smooth (implies no friction between board and platform).
**Diagram Description:**
* **Type:** Side view diagram showing a horizontal platform with a vertical drop-off.
* **Elements:**
* A horizontal surface labeled as a "光滑平台" (smooth platform). The surface has a hatched pattern below it indicating it is a solid structure.
* A point labeled "O" at the right edge of the platform.
* A horizontal rectangular bar representing the "均匀薄板" (uniform thin board) placed on the platform. Initially, the right end of the board is aligned with point O. The length of the board is labeled as 'l'.
* A small block is shown on the left end of the board.
* The platform ends at point O, and there is a vertical drop below O, followed by a horizontal hatched surface representing the ground or lower level. The vertical drop is shown as a stepped structure.
* **Labels and Annotations:**
* 'l': Length of the board.
* 'O': Point on the platform edge.
* **Implied context:** The block slides on the board, and both the block and board move horizontally on the smooth platform. After leaving the board at O, the block undergoes projectile motion. The board continues to move horizontally on the platform until its center reaches O.
解析
这道题主要考查了牛顿运动定律、匀变速直线运动规律、平抛运动以及动量守恒定律等知识点。
(1) 由题可知物块在薄板上做匀减速运动的加速度大小为 $a_1 = \mu g = 3m/s^2$
薄板的加速度$a_2$为: $a_2 = \frac{\mu mg}{m} = 3m/s^2$
对$l + \Delta l = v_0 t - \frac{1}{2}a_1 t^2$, $\Delta l = \frac{1}{2}a_2 t^2$
联立可得$v_0 = 4m/s$, $t = \frac{1}{3}s$
(2) 物块离开薄板得速度为$V_2$,
$v_2 = a_2 t = 1m/s$, 又物块离开薄板后薄板做匀速运动，物块做平抛运动，则当物块落到地面的时间为$t' = \frac{l}{v_2} = \frac{1}{3}s$!
则平台距地面的高度$h$为$h = \frac{1}{2}gt'^2 = \frac{5}{9}m$