An object initially at rest experiences an acceleration of 1.4 m/s^2 for 5.0s and then travels at that constant velocity for another 7.0s. What is the object's average velocity over the 12s interval?
Draw the time-velocity graph.
Initial velocity, u = 0
Acceleration, a = 1.4 m/s²
time, t = 5 seconds
Distance, S1 = ut+(1/2)at²
= 0*5 + (1/2)1.4*5²
= 17.5 m
Final velocity, v = u+at = 0+1.4*5 = 7 m/s
Distance, S2 = 7*7 = 49 m
Average velocity = (S1+S2)/(t1+t2)
=(17.5+49)/(5+7)
= 66.5/12
= 5.54 m/s
To find the object's average velocity over the 12-second interval, we need to calculate the total displacement and divide it by the total time.
First, let's calculate the displacement during the first 5 seconds when the object experiences an acceleration of 1.4 m/s^2.
Using the equation of motion:
displacement = initial velocity * time + 0.5 * acceleration * time^2
Since the object is initially at rest, the initial velocity is 0. Therefore:
displacement = 0.5 * acceleration * time^2
displacement = 0.5 * 1.4 * (5^2)
displacement = 0.5 * 1.4 * 25
displacement = 17.5 meters
After the acceleration phase, the object continues to travel at a constant velocity for another 7 seconds. Since the object's velocity remains constant, there is no further acceleration during this time. Therefore, the displacement during this time is given by:
displacement = velocity * time
Since the object maintains a constant velocity, we can use the velocity just after the acceleration phase, which is 1.4 m/s:
displacement = 1.4 * 7
displacement = 9.8 meters
Now, let's calculate the total displacement over the 12-second interval:
total displacement = displacement during acceleration + displacement during constant velocity
total displacement = 17.5 + 9.8
total displacement = 27.3 meters
Finally, we can calculate the average velocity:
average velocity = total displacement / total time
average velocity = 27.3 / 12
average velocity ≈ 2.28 m/s
Therefore, the object's average velocity over the 12-second interval is approximately 2.28 m/s.