In order to facilitate the first-year students to analyze such problems, it is suggested that the following questions should be clarified first:
1, the distance between two points on the number axis is the absolute value of the coordinate difference corresponding to these two points, that is, the difference between the number on the right and the number on the left MINUS. That is, the distance between two points on the number axis = the number represented by the right point-the number represented by the left point.
2. When a point moves on the number axis, because the right direction of the number axis is positive, the speed of moving to the right is regarded as positive speed, and the speed of moving to the right is regarded as negative speed. In this way, the coordinates of the point after moving can be directly obtained by adding the moving distance of the point on the basis of the starting point. That is, the number represented by point A is A, and the number represented by the left shift of unit B is A-B; After moving unit b to the right, the number is a+B.
3. The number axis is the product of the combination of numbers and shapes, and the motion analysis of points on the number axis should be combined with graphics. The path formed by the movement of points on the number axis can be regarded as the sum and difference relationship of line segments on the number axis.
Extended data:
Moving point problem in mathematics
1, the origin of a number axis is O, and the number corresponding to point A is-1 12, and point A translates evenly along the number axis through the origin to point B. ..
(1) If OA=OB, what is the number corresponding to point B? (2) It takes 3 seconds from point A to point B, and find the moving speed of this point.
(3) It takes 9 seconds to move from point A to point C at a uniform speed along the number axis, KC=KA, and the numbers corresponding to point K and point C are calculated respectively. ?
2. Moving point A moves from the origin to the negative direction of the number axis, and moving point B also moves from the origin to the positive direction of the number axis. After 3 seconds, the distance between two points is 15 unit lengths. It is known that the speed ratio of moving points A and B is 1: 4. (Speed unit: unit length/second)
(1) Find the speed of two moving points, and mark the positions of point A and point B when they move from the origin for 3 seconds on the number axis;
(2) If the two points A and B move from the position in (1) to the negative direction of the number axis at the same time, the origin will be right in the middle of the two moving points in a few seconds;
(3) In (2), when point A and point B continue to move in the negative direction of the number axis at the same time, another moving point C starts from the position of point B and moves to point A at the same time. When it meets point A, it immediately returns to point B, then moves to point A, and so on, until B catches up with point A, and C immediately stops moving. If point C keeps moving at a constant speed of 20 unit length/second, then point C starts to stop.
3. It is known that the numbers corresponding to two points A and B on the number axis are-1 and 3 respectively, and point P is the moving point on the number axis, and its corresponding number is X. ..
(1) If the distances from point P to point A and point B are equal, find the number corresponding to point P; ?
(2) Is there a point P on the number axis, so that the sum of the distances from point P to point A and point B is 6? If it exists, request the value of x; If it does not exist, explain the reasons;
(3) Point A and Point B move to the right at two unit lengths/min and 1 unit length/min respectively, while Point P moves from Point O to the left at six unit lengths/min. When encountering a, point P immediately moves to the right at the same speed, and keeps going back and forth between point A and point B to find the total distance that point P has traveled when point A and point B coincide.
4. The numbers corresponding to the two particles A and B on the number axis are -8 and 4. The two points A and B each move at a certain speed, and the speed of point A is 2 units/second.
(1) Point A and Point B start at the same time and meet at the origin to find the moving speed of Point B;
(2)A and B start at the same time at the speed of (1), and move in the positive direction of several axes, separated by 6 unit lengths in a few seconds;
(3)A and B start at the same time at the speed of (1) and move in the negative direction of several axes. At the same time, point C starts from the origin and moves in the same direction, and there is always CB: CA = 1: 2 in the process of movement. After a few seconds, C stays at-10, so ask.
5. On the number axis, the number represented by point A is -30, and the number represented by point B is 170.
(1) Find the number represented by the midpoint of A and B;
(2) One electronic frog M starts from point B and moves to the left at a speed of 4 units per second, while the other electronic frog N starts from point A and moves to the right at a speed of 6 units per second. Suppose they meet at point C, and find the number represented by point C;
(3) After the two electronic frogs meet at point C, they continue to move in the original direction. When electronic frog M is at point A, ask where electronic frog N is.
(4) If electronic frog M starts from point B and moves to the right at the same time, electronic frog N also moves to the right. Suppose they meet at point D, and find the number represented by point D. ..
6. It is known that there are three points A, B and C on the number axis, representing -24,-10 and 10 respectively. Two electronic ants, A and B, walk in opposite directions from points A and C, respectively. The speed of A is 4 units per second.
(1) How many seconds later, the sum of the distances from A to A, B and C is 40 units?
(2) If the speed of B is 6 units per second, then two electronic ants, A and B, walk in opposite directions from two points, A and C, respectively. At what point on the number axis do A and B intersect?
(3) Under the conditions of (1) and (2), when the sum of the distances from A to A, B and C is 40 units, A turns around and returns. Can A and B still meet on the number axis? If yes, find the meeting point; If not, please explain why.
7. It is known that the numbers corresponding to points A and B on the number axis are-1 and 3 respectively, and point P is the moving point on the number axis, and its corresponding number is X. ..
(1) If the distances from point P to point A and point B are equal, find the number corresponding to point P;
(2) Is there a point P on the number axis, so that the sum of the distances from point P to point A and point B is 5? If it exists, request the value of x, if it does not exist, please explain why?
(3) When point P moves from point O to the left at a speed of one unit length per minute, point A moves to the left at a speed of five unit lengths per minute, and point B moves to the left at a speed of twenty unit lengths per minute. Ask them to start at the same time. After a few minutes, are the distances from point P to point A and point B equal?