Question Part
Points
Submissions Used
1 2 3 4 5
0/5 –/5 5/5 0/5 0/5
1/10 0/10 1/10 2/10 2/10
Total
5/25
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 .
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.
The response you submitted has the wrong sign. m/s2
The response you submitted has the wrong sign.
To find the answers to the questions, we need to use some basic concepts of circular motion and vector mathematics. Let's go through each question step by step:
(a) To find the acceleration at point B, we need to calculate the change in velocity over time. Since the car is traveling in a circular path, the change in velocity will be the change in its direction.
1. First, we need to find the magnitude of the change in velocity. This is equal to the speed of the car multiplied by the change in angle.
Speed of the car = arc length / time taken = 210 m / 38.0 s = 5.526 m/s
Change in angle = 35.0°
2. Next, we need to find the direction of the change in velocity. In this case, it is perpendicular to the radius of the circular path and points towards the center of the circle (centripetal direction).
This direction can be represented by the unit vector pointing towards the center, -r̂ (negative because it is in the opposite direction of the positive r̂).
3. Finally, we can calculate the change in velocity vector.
Change in velocity = (Speed of the car) * sin(Change in angle) * -r̂
= 5.526 m/s * sin(35.0°) * -r̂
= -3.117 m/s * -r̂
= 3.117 m/s * r̂ (We drop the negative sign as it was just indicating direction)
So, the acceleration when the car is at B is 3.117 m/s² * r̂.
(b) To determine the car's average speed, we can use the formula:
Average speed = Total distance traveled / Total time taken
In this case, the distance traveled is the length of arc ABC, which is 210 m, and the total time taken is 38.0 s.
Average speed = 210 m / 38.0 s = 5.526 m/s
So, the car's average speed is 5.526 m/s.
(c) To determine the car's average acceleration during the 38.0 s interval, we need to calculate the change in velocity over the time interval.
1. Since the car is traveling in a circular path at a constant speed, the magnitude of its acceleration is equal to zero. This means there is no change in speed.
2. However, there is a change in direction, which is represented by the change in velocity vector. As we calculated earlier, the magnitude of the change in velocity vector is 3.117 m/s, and it is pointing towards the center of the circle.
So, the car's average acceleration during the 38.0 s interval is 3.117 m/s² * r̂.
Note: The negative sign in your submitted response indicates the direction, but the question specifically asks for the answer in terms of unit vectors r̂ and θ̂.