AP > AP物理C卷：运动学方程

2021-09-17
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**运动学方程 ▉**

**Kinematic equations**

Kinematics is the study of motion under constant acceleration. It allows us to predict the motion of objects given some initial conditions. Specifically, it relates all of the motion’s variables in three simple kinematic equations. These variables are summarized in the table below

All these kinematic variables with the exception of are vector quantities which means they can be negative. A missing negative sign is usually responsible for many errors when employing kinematic equations. If a coordinate system is picked where the upward direction is positive, then the acceleration due to gravity which points downwards must take a negative value. Conversely, if you pick the downward direction to be positive, then the acceleration due to gravity is a positive number. It does not matter how you pick your coordinate system, what matters is that you are consistent which your choice.

Deriving the kinematic equations

Bear in mind that kinematics deals with motion under constant acceleration; this implies that there is no change in acceleration,

In this case, the average acceleration of an object is just the acceleration itself. This acceleration will be given by the rate at which the velocity changes per unit of time. Mathematically, this rate is:

where the subscripts f and i represent the final and the initial states, respectively. Solving for the final velocity

This is the first kinematic equation. It relates velocities and acceleration in a given time interval. As you can see, it doesn’t provide any information about the object’s position or displacement.

Consider an object with initial velocity

at initial time

then at a later time

the objects velocity will be

This situation is represented in the graph below (velocity vs time graph)

We know that the displacement

of an object which is traveling at constant velocity

in a time interval

is given by

Notice that this expression is the area of the rectangle in the graph above its base is

and whose height is

Finding the displacement

when the velocity is not constant implies accounting for the area of the triangle that is stacked above the rectangle. From the graph we can see that the height of this triangle is

Combining both areas together gives the following expression:

The first kinematic equation gives us an expression for which we can substitute into the expression above giving

This is the second kinematic equation. Note that this equation relates position and displacement to the other variables. Note however, that so far both kinematic equations contain the time interval

Sometimes motion occurs over unknown time intervals so it’s convenient to have another kinematic equation that deals with these situations.

The final kinematic equation is obtained by combining the first two; we take the first kinematic equation and solve for

as follows

Then, we substitute this expression into the second kinematic equation giving:

This is the third kinematic equation.

The three kinematic equations summarize the relationships between all variables of motion. In solving problems of motion, it is a good idea to list all your known and unknown quantities so that you can select the appropriate kinematic equation. In more involved problems, you may have to work with more than one kinematic equation. Never use a kinematic equation if the acceleration is not constant.

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