A body of mass m moving along a straight line covers half the distance with a speed of 2 ms. The remaining half of the distance is covered in two equal time intervals with a speed of 3 ms¹ and 5 ms¹ respectively. The average speed of the particle for the entire journey

视频信息

答案文本 复制

视频字幕 复制

Let's analyze this motion problem step by step. A body moves along a straight line with varying speeds. The journey is divided into segments: the first half of the total distance is covered at a constant speed of 2 meters per second. The remaining half distance is covered in two equal time intervals - first at 3 meters per second, then at 5 meters per second. Our goal is to find the average speed for the entire journey. Let's analyze the first half of the journey in detail. For this segment, the body travels a distance equal to d over 2, where d is the total distance of the journey. The speed during this segment is constant at 2 meters per second. Using the fundamental relationship that time equals distance divided by speed, we can calculate the time for this segment. Time t1 equals d over 2 divided by 2, which simplifies to d over 4. This establishes our first time interval for the journey. Now let's examine the constraints for the second half of the journey. The remaining distance of d over 2 must be covered in two equal time intervals, each lasting time t. In the first interval, traveling at 3 meters per second for time t covers a distance of 3t. In the second interval, traveling at 5 meters per second for time t covers a distance of 5t. Since these two distances must sum to the remaining distance d over 2, we have the constraint equation: 3t plus 5t equals d over 2. This simplifies to 8t equals d over 2, giving us t equals d over 16. Let's calculate all the time intervals and verify our solution. For the first half, we found t1 equals d over 4. For the second half, each equal time interval is t equals d over 16. The total time for the second half is 2t, which equals 2 times d over 16, or d over 8. Let's verify our distances are correct: the first interval covers 3 times d over 16 equals 3d over 16, and the second interval covers 5 times d over 16 equals 5d over 16. Adding these gives 8d over 16, which simplifies to d over 2, confirming our constraint is satisfied. The total journey time is t1 plus 2t, which equals d over 4 plus d over 8, giving us 3d over 8. Now we can calculate the average speed for the entire journey using the fundamental formula: average speed equals total distance divided by total time. We have established that the total distance is d and the total time is 3d over 8. Substituting these values, average speed equals d divided by 3d over 8. This is equivalent to d multiplied by 8 over 3d. The d terms cancel out, giving us 8 over 3 meters per second. Converting to decimal form, this equals approximately 2.67 meters per second. Therefore, the average speed of the particle for the entire journey is 8 over 3 meters per second, or 2.67 meters per second.