An automobile accelerates from rest at 1.3m/s^2 for 19s. The speed is then held constant for 23s, after which there is an acceleration of -0.9m/s^2 until the automobile stops.
What total distance was traveled?
Answer in units of km.
X1 = (1/2)a*19^2 = 234.7 m
Attained velocity:
Vmax = 1.3*19 = 24.7 m/s
X2 = 24.7 * 23 s = 568.1 m
X3 = (average speed)*(deceleration time)
= (Vmax/2)*(Vmax/a)= ___?
Add them up for total distance, and convert to km.
here is another way:
make it to X1,X2,X3
X1=(1/2)(1.3*19)(19)
X2=(1.3*19)(23)
X3=(1.3*19)/(0.9)(1/2)(1.3*19)
total distance = X1+X2+X3
To find the total distance traveled, we need to calculate the distance traveled during each phase of the motion and then add them together.
First, let's calculate the distance during the acceleration phase from rest. We can use the formula:
distance = (initial velocity * time) + (0.5 * acceleration * time^2)
Since the initial velocity is 0 (since it started from rest) and the acceleration is 1.3 m/s^2, and the time is 19 seconds, we can substitute these values into the formula:
distance1 = (0 * 19) + (0.5 * 1.3 * 19^2)
distance1 = 0 + (0.5 * 1.3 * 19^2)
distance1 = 0 + (0.5 * 1.3 * 361)
distance1 = 0 + (0.5 * 469.3)
distance1 = 0 + 234.65
distance1 = 234.65 meters
Next, let's calculate the distance during the constant speed phase. The speed is held constant for 23 seconds, so the distance traveled during this phase can be calculated by multiplying the speed by the time:
distance2 = speed * time
Since the speed is constant, we can use the speed during the acceleration phase, which is the final speed after accelerating:
distance2 = (1.3 * 19)
distance2 = 24.7 meters
Finally, let's calculate the distance during the deceleration phase until the automobile stops. We can use the same formula we used for the first phase:
distance3 = (initial velocity * time) + (0.5 * acceleration * time^2)
Since the final speed is 0 (the automobile stops), the initial velocity for this phase is the speed during the constant speed phase (24.7 m/s) and the acceleration is -0.9 m/s^2. The time is unknown, but we know that the automobile comes to a stop during this phase. To find the time, we can use the formula:
final speed = initial velocity + (acceleration * time)
0 = 24.7 + (-0.9 * time)
-24.7 = -0.9 * time
time = -24.7 / -0.9
time ≈ 27.4 seconds
Now, we can substitute these values into the formula:
distance3 = (24.7 * 27.4) + (0.5 * -0.9 * 27.4^2)
distance3 = 677.78 + (-0.5 * 0.9 * 752.76)
distance3 = 677.78 + (-0.5 * 677.48)
distance3 = 677.78 + (-338.74)
distance3 ≈ 339.04 meters
Finally, we can find the total distance traveled by adding the distances calculated during each phase:
total distance = distance1 + distance2 + distance3
total distance = 234.65 + 24.7 + 339.04
total distance ≈ 598.39 meters
To convert this to kilometers, we divide by 1000 (since there are 1000 meters in a kilometer):
total distance = 598.39 / 1000
total distance ≈ 0.59839 kilometers
Therefore, the total distance traveled is approximately 0.59839 kilometers.