Constant Acceleration Motion匀加速运动
Constant acceleration motion can be characterized by formulae and by motion graphs.
Show development of formuli from calculus.从微积分推导公式
匀加速运动可以用公式和运动图象来描述。 |
Index Motion concepts 索引运动概念 | ||
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Calculus Application for Constant Acceleration微积分在恒定加速度中的应用
The motion equations for the case of constant acceleration can be developed by integration of the acceleration. The process can be reversed by taking successive derivatives.
On the left hand side above, the constant acceleration is integrated to obtain the velocity. For this indefinite integral, there is a constant of integration. But in this physical case, the constant of integration has a very definite meaning and can be determined as an intial condition on the movement. Note that if you set t=0, then v = v0, the initial value of the velocity. Likewise the further integration of the velocity to get an expression for the position gives a constant of integration. Checking the case where t=0 shows us that the constant of integration is the initial position x0. It is true as a general property that when you integrate a second derivative of a quantity to get an expression for the quantity, you will have to provide the values of two constants of integration. In this case their specific meanings are the initial conditions on the distance and velocity. 在上方的左边，恒定加速度被积分以得到速度。对于这个不定积分，存在一个积分常数。但在这种物理情况下，积分常数具有非常明确的含义，并可以确定为运动的初始条件。注意，如果将t设为0，则v = v₀，即速度的初始值。同样，对速度进行进一步积分以得到位置的表达式时，也会出现一个积分常数。检查t=0的情况，可以看到积分常数就是初始位置x₀。正如一般性质所指出的，当你对一个量的二阶导数进行积分以得到该量的表达式时，必须提供两个积分常数的值。在这种情况下，这两个常数的具体含义是距离和速度的初始条件。
Show framework for time dependent acceleration.时间依赖加速度的框架
在恒定加速度情况下的运动方程可以通过对加速度进行积分来推导。这个过程也可以通过逐次求导来逆向进行。 |
Index Motion concepts Velocity and acceleration 索引 运动概念 速度与加速度 | ||
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Time Dependent Acceleration时间依赖的加速度
If the acceleration of an object is time dependent, then calculus methods are required for motion analysis. The relationships between position, velocity and acceleration can be expressed in terms of derivatives or integrals.  Show constant acceleration case.展示恒定加速度情况。
Acceleration as polynomial in time.加速度作为时间的多项式
如果物体的加速度随时间变化，那么为了分析运动就需要使用微积分方法。位置、速度和加速度之间的关系可以用导数或积分来表示。 |
Index Motion concepts Velocity and acceleration 索引 运动概念 速度与加速度 | ||
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