a bus moves from rest with a uniform acceleration of 2ms² for the first 10 seconds. it then accelerate at a uniform rate of 1ms² for another 15seconds. it continues at a constant speed for 70seconds and finally came to rest in 20seconds by uniform acceleration, calculate the average speed for the whole journey.
To solve this problem, we need to break down the motion of the bus into different segments and calculate the distance covered in each segment.
1. First 10 seconds: The bus accelerates uniformly with an acceleration of 2 m/s². We can use the equation of motion: s = ut + (1/2)at², where s is the distance covered, u is the initial velocity, a is the acceleration, and t is the time. In this case, u = 0 (since the bus starts from rest), a = 2 m/s², and t = 10 seconds.
Using the equation, s = 0 + (1/2)(2)(10)², we get s = 100 meters.
2. Next 15 seconds: The bus accelerates uniformly, but with a different acceleration of 1 m/s². Using the same equation of motion, s = ut + (1/2)at², where u = initial velocity, a = acceleration, and t = time. Here, u = initial velocity = final velocity from the previous segment = 20 m/s (as the bus continues at constant speed). a = 1 m/s², and t = 15 seconds.
Using the equation, s = 20(15) + (1/2)(1)(15)², we get s = 450 + 112.5 = 562.5 meters.
3. Next 70 seconds: The bus continues at a constant speed, which means there is no acceleration. Therefore, we can use the equation s = ut, where u = initial velocity and t = time. Here, u = final velocity from the previous segment = 20 m/s, and t = 70 seconds.
Using the equation, s = 20(70) = 1400 meters.
4. Final 20 seconds: The bus decelerates (accelerates negatively) uniformly to come to rest. The initial velocity is the final velocity from the previous segment = 20 m/s. The acceleration is negative because it opposes the motion. Let's call it -a. The time is 20 seconds.
Using the equation, s = ut + (1/2)at², where a = -1 m/s² (negative since it's deceleration) and t = 20 seconds, we get s = 20(20) + (1/2)(-1)(20)² = 400 - 200 = 200 meters.
Now, to find the total distance covered, we sum up the distances from each segment: 100 + 562.5 + 1400 + 200 = 2262.5 meters.
Finally, we calculate the average speed using the formula: Average speed = total distance covered / total time taken.
The total time taken is the sum of the times from each segment: 10 + 15 + 70 + 20 = 115 seconds.
Average speed = 2262.5 meters / 115 seconds ≈ 19.67 m/s (rounded to two decimal places).
Therefore, the average speed for the whole journey is approximately 19.67 m/s.
2m/s2 And 16m/s
20m
since
v = at and
s = vt + 1/2 at^2 for each section,
total distance: (1/2 * 2 * 10^2) + (2*10*15 + 1/2 * 1 * 15^2) + (2*10 + 1*15)*70 + (2*10 + 1*15)*20 - 1/2 (35/20) * 20^2
avg speed is then totaldistance / totaltime