A car initially traveling eastward turns north by traveling in a circular path at uniform speed as in the figure below. The length of the arc ABC is 210 m, and the car completes the turn in 38.0 s.
(a) What is the acceleration when the car is at B located at an angle of 35.0°? Express your answer in terms of the unit vectors and .
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The response you submitted has the wrong sign. m/s2 + m/s2
(b) Determine the car's average speed.
. m/s
(c) Determine its average acceleration during the 38.0 s interval.
To answer question (a), we need to find the acceleration of the car when it is at point B.
First, let's calculate the velocity of the car at point B using the information given. We are given the length of the arc AB (210 m) and the time it takes to complete the turn (38.0 s).
The velocity is calculated by dividing the distance traveled (length of arc AB) by the time taken:
velocity = distance / time
v = 210 m / 38.0 s
v ≈ 5.53 m/s
Now, we need to find the change in velocity Δv when the car is at point B. This can be calculated by considering the change in direction as the car turns:
Δv = v - (-v)
Δv = 2v
Δv ≈ 2 * 5.53 m/s
Δv ≈ 11.06 m/s
Next, we need to determine the change in position Δs when the car is at point B. Since the car has turned in a circular path, the change in position can be calculated as the length of the arc BC:
Δs = length of arc BC = 210 m
Finally, we can find the acceleration at point B using the equation:
acceleration = Δv / Δt
a = Δv / Δt
We have already calculated Δv as 11.06 m/s, and the time Δt is not provided in the question. However, since the car completes the turn in 38.0 s, we can assume that the time taken to reach point B is half of the total time:
Δt = 38.0 s / 2
Δt = 19.0 s
Now we can calculate the acceleration:
a = 11.06 m/s / 19.0 s
a ≈ 0.582 m/s²
Therefore, the acceleration of the car at point B is approximately 0.582 m/s² in the direction of the unit vector.
Moving on to question (b), to find the car's average speed we need to divide the total distance traveled by the total time taken.
The total distance traveled is the length of the arc ABC, given as 210 m. And the total time taken to complete the turn is given as 38.0 s.
Average speed = total distance / total time
Average speed = 210 m / 38.0 s
Average speed ≈ 5.53 m/s
Therefore, the car's average speed is approximately 5.53 m/s.
Finally, for question (c), to find the average acceleration during the 38.0 s interval, we can use the formula:
average acceleration = change in velocity / time interval
We have already calculated the change in velocity Δv as 11.06 m/s, and the time interval Δt is given as 38.0 s.
average acceleration = 11.06 m/s / 38.0 s
average acceleration ≈ 0.291 m/s²
Therefore, the car's average acceleration during the 38.0 s interval is approximately 0.291 m/s².