At one instant a bicyclist is 40 m due east of a park's flagpole, going due south with a speed of 18 m/s. Then, 23 s later, the cyclist is 55 m due north of the flagpole, going due east with a speed of 12 m/s. For the cyclist in this 23 s interval, find each of the following.
(a) displacement
magnitude m
direction ° north of west
(b) average velocity
magnitude m/s
direction ° north of west
(c) average acceleration
magnitude m/s2
direction ° north of east
(a) The displacement of the cyclist can be found by subtracting the initial position from the final position. The initial position is 40 m due east of the flagpole, and the final position is 55 m due north of the flagpole. So, the displacement is the hypotenuse of a right triangle with legs of 40 m and 55 m. We can use the Pythagorean theorem to find the magnitude of the displacement:
displacement magnitude = √(40² + 55²) ≈ 68.83 m.
To find the direction, we can use trigonometry. The angle θ can be found by taking the inverse tangent of the opposite side divided by the adjacent side:
θ = tan^(-1)(55/40) ≈ 54.47°.
So, the direction of the displacement is 54.47° north of east.
(b) The average velocity is given by the displacement divided by the time interval. The displacement is approximately 68.83 m and the time interval is 23 s. So, the average velocity is:
average velocity magnitude = displacement / time interval = 68.83 / 23 ≈ 2.99 m/s.
The direction of the average velocity is the same as the direction of the displacement, which is 54.47° north of east.
(c) To find the average acceleration, we need to know the change in velocity and the time interval. The change in velocity can be found by subtracting the initial velocity from the final velocity. The initial velocity is -18 m/s (southward) and the final velocity is 12 m/s (eastward). So, the change in velocity is:
change in velocity = final velocity - initial velocity = 12 - (-18) = 30 m/s.
The average acceleration is given by the change in velocity divided by the time interval. The change in velocity is 30 m/s and the time interval is 23 s. So, the average acceleration is:
average acceleration magnitude = change in velocity / time interval = 30 / 23 ≈ 1.30 m/s².
The direction of the average acceleration is not specified in the question, so we can't determine it using the given information.
To solve this problem, we need to break it down into steps and use the kinematic equations of motion.
Step 1: Determine the displacement of the cyclist.
The displacement is the change in position of the cyclist from the initial position to the final position. In this case, the cyclist starts at a point 40 m east of the flagpole and ends at a point 55 m north of the flagpole.
To find the displacement, we use the Pythagorean theorem:
Displacement = √(40^2 + 55^2) = √(1600 + 3025) = √(4625) = 67.96 m
So, the magnitude of the displacement is 67.96 m.
To find the direction, we use trigonometry:
tan(θ) = (opposite/adjacent) = (55/40)
θ = tan^(-1)(55/40) = 54.46°
Therefore, the direction is 54.46° north of east. However, since we want the direction ° north of west, we subtract this angle from 90°:
Direction = 90° - 54.46° = 35.54° north of west.
So, the displacement has a magnitude of 67.96 m and a direction of 35.54° north of west.
Step 2: Determine the average velocity of the cyclist.
Average velocity is defined as the displacement divided by the time interval. In this case, the time interval is 23 seconds.
Average Velocity = Displacement / Time Interval = 67.96 m / 23 s = 2.956 m/s
To find the direction, we use the same angle from the displacement calculation: 35.54° north of west.
So, the average velocity has a magnitude of 2.956 m/s and a direction of 35.54° north of west.
Step 3: Determine the average acceleration of the cyclist.
In this problem, we were not provided with the time intervals within the 23 second time frame. Without this information, we cannot calculate the average acceleration. Average acceleration is defined as the change in velocity divided by the time interval.
Therefore, we cannot determine the magnitude or direction of the average acceleration.
In summary:
(a) Displacement: Magnitude = 67.96 m, Direction = 35.54° north of west.
(b) Average Velocity: Magnitude = 2.956 m/s, Direction = 35.54° north of west.
(c) Average Acceleration: Not enough information provided to calculate the magnitude or direction.