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.
(a) To find the acceleration when the car is at B, we need to consider the changes in velocity. Since the car is traveling in a circular path, its velocity is constantly changing, and we can find the acceleration using the centripetal acceleration formula.
The centripetal acceleration is given by:
ac = v^2 / r
Where v is the velocity of the car at point B and r is the radius of the circular path.
To find v at point B, we first need to find the angular velocity (ω) using the formula:
θ = ωt
Given that the car completes the turn in 38.0 s and the arc length ABC is 210 m, we can calculate the angle:
θ = arc length / radius = 210 m / r
Now we can solve for ω:
θ = ωt
35.0° = ω * 38.0s
Solving for ω, we get:
ω = 35.0° / 38.0s = 0.921°/s
Now, we can use the angular velocity (ω) to find the linear velocity (v) at point B:
v = ω * r
Since we don't know the radius (r), we need to find it using the given arc length (210 m):
arc length = radius * angle
210 m = r * 35.0°
Solving for r, we get:
r = 210 m / 35.0° = 6.00 m
Finally, we can calculate the linear velocity at point B:
v = ω * r
v = (0.921°/s) * 6.00 m = 5.53 m/s
Now that we have the linear velocity (v) at point B, we can find the acceleration (a) using the centripetal acceleration formula:
ac = v^2 / r
Substituting the values, we get:
ac = (5.53 m/s)^2 / 6.00 m = 5.08 m/s^2
Therefore, the acceleration when the car is at B is 5.08 m/s^2 in the direction of the center of the circular path.
(b) The average speed of an object is defined as the total distance traveled divided by the total time taken. In this case, the car completes a circular path with a length of 210 m in 38.0 s.
Therefore, the average speed is:
Average speed = Total distance / Total time = 210 m / 38.0 s
Evaluating the expression, we get:
Average speed = 5.53 m/s
Hence, the car's average speed is 5.53 m/s.
(c) The average acceleration of an object is defined as the change in velocity divided by the time taken.
In this case, the car starts and ends at the same point, so the change in velocity is zero. Therefore, the average acceleration during the 38.0 s interval is zero m/s^2.