III.
(9) Change of displacement The transformation law
(1) During any equal time interval, the horizontal displacement remains unchanged, and Δx=v0Δt.
(2) During any equal time interval Δt, the displacement difference in the vertical direction remains unchanged, that is,
(10) The speed of the two partial motions and the combined motion speed at any moment form a vector right triangle.
Example 1. Two small balls are thrown at the same time in the horizontal direction from the same point in the air. Their initial velocity direction is opposite, and their sizes are
and
. Find how long it takes to get the angle between the speed of the two small balls to be 90°?
parsing: Suppose the time after the two small balls are thrown, the angle between their speed is 90°, and the angle with the vertical direction is divided into
, respectively, α and β, respectively, for the two small balls, respectively. Build a speed vector right-angle triangle, as shown in the figure, according to the figure, you can get:
①
①
and because
�
(11) The displacement of the two partial motions and the displacement of the combined motions form a vector right-angle triangle in any period of time.
Example 2. As shown in Figure A, the small balls a and b are thrown horizontally from the vertex of the triangle inclined surface at an initial velocity of equal size and opposite directions. It is known that the inclination angles of the two inclined surfaces are
and
. Find the ratio of the time it takes for the small balls a and b to fall on the inclined surface? (Suppose the triangle inclined surface is long enough)
parsing: According to the recommendation On making the right-angle triangle at this time, as shown in Figure B,
pair a has:
①
①
versus b has:
②
6 versus b has:
②
6 .
Inference 6 proves that at any position of the object that is flat-throwing, the angle between the end velocity direction and the horizontal direction is θ, and the angle between displacement and horizontal is
, then tanθ=2tan
.
proof: As shown in the figure, the flat casting motion law is
,
,
so
.
Example 3. As shown in the figure, the top of a fixed inclined surface with an object with a self-inclination angle of θ is thrown in the horizontal direction and falls on the inclined surface. The angle between the velocity and the horizontal direction when the object contacts the inclined surface
satisfies ( )
A,
B,
C,
C,
D,
D,
66 parsing: Based on the inference 6, we can see that the correct option is D.
Inference 7 proves that the reverse extension line of the object that is doing flat-throwing motion at any moment must pass through the midpoint of the horizontal displacement at this time.
proof: As shown in the figure, B is the midpoint of OA. Suppose the initial velocity of the flat-cast object is
. The time from origin O to point A is t, and the coordinate of point A is
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , the coordinates of point B are
, then
,
,
.
, and the solution is
. That is, the intersection point B of the reverse extension line in the velocity direction of the last state and the x-axis must be the midpoint of the horizontal displacement OA at this moment.
Example 4. As shown in the figure, a small ball is thrown from the coordinate origin along the horizontal axis Ox at the speed of
2m/s. After a period of time, M is P point on the Ox axis. The projection of the sphere is used as the tangent line of the sphere trajectory at point P and extends inversely, intersecting with the Ox axis at point Q. It is known that QM=3m, so what is the time for the sphere to move?
Analysis: From the inference 7, it can be seen that Q is the midpoint of OM, so during the movement from point O to point P, the horizontal displacement s level of the sphere occurs = OM=2QM=6m. Since the horizontal direction is moving in a uniform linear manner, the time the ball moves during this process is t=3s.
4. Typical examples of flat casting movement
Source: This article was edited and organized by the high school physics learning station. The key is sharing. The copyright belongs to the original author's original source.
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