A particle accelerates from rest at 2.0 m/s2 for 2.0 s, then at 1.0 m/s2 for 3.0 s, then at constant velocity for 4.0 s. Sketch a graph of velocity versus time and determine (a) the average acceleration during the entire motion, and (b) the displacement.
follow the directions.
the slope of the velocity/time graph at any point is acceleration.
The cumumative area under the graph is displacement.
To sketch the graph of velocity versus time, we need to break down the motion into different intervals and calculate the velocity at each interval.
1. From rest to 2.0s: The particle accelerates at 2.0 m/s² for 2.0 seconds. We can use the equation:
v = u + at
where v is the final velocity, u is the initial velocity (0 m/s), a is the acceleration (2.0 m/s²), and t is the time (2.0 s).
v = 0 + (2.0 m/s²)(2.0 s) = 4.0 m/s
So, at t = 2.0 s, the velocity is 4.0 m/s.
2. From 2.0s to 5.0s: The particle accelerates at 1.0 m/s² for 3.0 seconds. Again, we can use the same equation:
v = u + at
where u is the final velocity from the previous interval (4.0 m/s), a is the new acceleration (1.0 m/s²), and t is the time (3.0 s).
v = 4.0 m/s + (1.0 m/s²)(3.0 s) = 7.0 m/s
So, at t = 5.0 s, the velocity is 7.0 m/s.
3. From 5.0s onwards: The particle moves at constant velocity. Therefore, the velocity remains at 7.0 m/s.
Now we can plot the values on a graph. The y-axis represents velocity, and the x-axis represents time.
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| *
8 | *-----------------------------*
| * *
6 | * *
| * *
4 | * *
| * *
2 |* *
|_______________________________>
0 2 4 5 6 7 8 10 12 14
(a) To find the average acceleration during the entire motion, we need to calculate the total change in velocity divided by the total time elapsed. In this case, the total change in velocity is 7.0 m/s - 0 m/s = 7.0 m/s, and the total time elapsed is 8.0 seconds (2.0 s + 3.0 s + 4.0 s).
average acceleration = (total change in velocity) / (total time elapsed)
= 7.0 m/s / 8.0 s
= 0.875 m/s²
So, the average acceleration during the entire motion is 0.875 m/s².
(b) To find the displacement, we need to calculate the area under the graph. Since the velocity remains constant at 7.0 m/s for 4.0 seconds, the displacement during that interval is:
displacement = velocity × time
= 7.0 m/s × 4.0 s
= 28.0 m
Therefore, the displacement is 28.0 meters.