Motion with time dependent acceleration
时间依赖加速度的运动
Given an object which starts from position x=0 and moves along a straight line with position given by
给定一个从位置x=0出发，沿直线运动的物体，其位置由
find:
a. The expression for velocity and the time at which the velocity is zero.
b. The expression for acceleration and the time at which the acceleration is zero.
c. The average velocity over the first 3 seconds of motion.
d. Show graphs of the position, velocity and acceleration for the first 4 seconds of motion.
Solution:
解法：
a. The velocity is given by the derivative of the position.
 | and for v=0 the time is
并且当v=0时，时间是 |  . |
a. 速度由位置的导数给出
b. The acceleration is given by the derivative of the velocity.
 | and for a=0 the time is t=0.
当a=0时，时间是t=0。 |
b. 加速度由速度的导数给出
c. The average velocity must be calculated from the displacement divided by the time.
| (Caution! The expression
（注意！表达式 |  |
| is not valid for this case since the acceleration is not constant.)
这种情况下不适用，因为加速度不是恒定的。 |
平均速度必须通过位移除以时间来计算。
The average velocity for the first 3 seconds of motion is given by
 | where
其中 |  |
so the average velocity is -25 m/s. You can check to see that the constant acceleration expression for average velocity gives the erroneous value -22.5 m/s.
前3秒内的平均速度由下式给出，因此平均速度为-25 m/s。你可以检查一下，恒定加速度情况下平均速度的表达式给出的错误值为-22.5 m/s。
d. Motion graphs.
d. 运动图线。
Motion equations as polynomials in time.
作为时间的多项式运动方程
| Example problems in mechanics
力学例题 |
求：a. 速度的表达式以及速度为零的时间。b. 加速度的表达式以及加速度为零的时间。c. 运动前3秒的平均速度。d. 绘制前4秒内位置、速度和加速度的图像。
Polynomial derivatives
索引多项式导数
力学
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