A runner circles a track of radius 50 m in 50 s moving at a constant rate. If the runner was initially moving north, what has been the runner's average acceleration when halfway around the track?
First get the speed, which is constant.
It is 2 pi*50/50 = 6.28 m/s
The DIRECTION of velocity changes 180 degrees in the time it take to go half way around the track, 25 s. The velcoity changes by 12.56 m/s, from 6.28 to -6.28.
The AVERAGE acceleration is the change in the velocity vector divided by the time, which is
a(av) = 2*6.28/25 = 0.502 m/s^2
The value of the acceleration scalar is always V^2/R = 0.789 m/s^2, but its direction is constantly changing. I don't believe that is what they area after.
Well, if the runner is running in circles around the track, I'd say their average acceleration when halfway around the track is... confused! You see, acceleration refers to a change in velocity, and halfway around the track, the runner is basically back where they started. So, their velocity hasn't really changed much, if at all. So, if they're at the same speed, I guess we could say their average acceleration is "zero ducks given." It's like they're in a perpetual loop-de-loop of non-acceleration hilarity!
To find the average acceleration of the runner halfway around the track, we need to calculate the change in velocity and the time taken. The average acceleration is given by the formula:
Average acceleration = (Change in velocity) / (Time taken)
Since the runner is moving at a constant rate, we can calculate the change in velocity by subtracting the initial velocity from the final velocity.
To find the initial velocity, we need to know the distance covered by the runner halfway around the track.
The distance halfway around the track can be calculated using the formula for the circumference of a circle:
Circumference = 2 * π * radius
Given that the radius of the track is 50 m, the distance halfway around the track is:
Distance halfway = 2 * π * 50 = 100π m
Since the runner takes 50 seconds to complete one full lap, the time taken to cover half the distance is:
Time taken halfway = 50 s / 2 = 25 s
Now, we know the distance covered and the time taken, so we can calculate the average speed using the formula:
Average speed = Distance / Time
Average speed halfway = (100π) m / 25 s = 4π m/s
Since the runner is moving at a constant rate, the average speed halfway around the track is also the final velocity.
Now, we can calculate the initial velocity by considering the given information that the runner started moving north. Since there is no change in the north direction, the initial velocity in the north direction is 0 m/s.
Therefore, the change in velocity is:
Change in velocity = Final velocity - Initial velocity
Change in velocity = 4π m/s - 0 m/s
Change in velocity = 4π m/s
Finally, we can calculate the average acceleration:
Average acceleration = Change in velocity / Time taken
Average acceleration = (4π m/s) / (25 s)
Average acceleration = (4π/25) m/s²
So, the runner's average acceleration when halfway around the track is (4π/25) m/s².