Suppose runner A starts at the 0 degree mark on a circular track (circumference = 400m). And let runner B start at the 180 degree mark. If runner A starts running counter clockwise with acceleration 2m/s^2, and runner B runs clockwise with constant speed of 7 m/s, at what angle do they meet?
To find the angle at which runners A and B meet, we need to calculate the time it takes for them to cover the respective distances and then convert that time into an angle.
First, let's find the distance traveled by each runner until they meet. The track's circumference is 400m, so runner A needs to cover a distance of 200m (half the track) to reach runner B.
Next, let's find the time it takes for runner A to cover this distance. We can use the kinematic equation:
s = ut + (1/2)at^2
where:
s = distance
u = initial velocity
a = acceleration
t = time
Since the initial velocity of runner A is 0m/s, the equation simplifies to:
s = (1/2)at^2
Substituting the known values:
200 = (1/2)(2)t^2
Simplifying the equation:
t^2 = 200/2 = 100
t = √100 = 10 seconds
Now that we know the time it takes for runner A to cover the distance, let's find the angle covered in that time.
The formula to convert time into an angle on a circular track is:
angle = (time / total time) * 360 degrees
The total time for one complete revolution on the track is given by the formula:
total time = (circumference / speed)
For runner B, the total time is:
total time = 400m / 7m/s = 57.14 seconds (rounded to 2 decimal places)
Now we can calculate the angle runner A covers:
angle = (10s / 57.14s) * 360 degrees
angle = 0.175 * 360 degrees
angle ≈ 63.00 degrees
Therefore, runners A and B will meet at approximately the 63.00 degree mark on the circular track.
To determine at what angle runners A and B will meet, we need to find the time it takes for them to meet on the circular track.
Let's start by calculating the time it takes for runner A to reach the meeting point. We can use the equation of motion:
s = ut + (1/2)at^2
Where s is the displacement, u is the initial velocity, a is the acceleration, and t is the time.
For runner A:
Initial velocity, u = 0
Acceleration, a = 2 m/s^2
Displacement, s = 180 degrees (half the circle, from starting point to meeting point)
We know that the circumference of the track is 400m, meaning the full circle is 360 degrees. So, the displacement in meters is given by:
s = (180/360) * 400 = 200m
Using the equation of motion, we can rearrange it to solve for time:
t = √(2s/a)
t = √(2 * 200 / 2) = √200 = 14.14 seconds (approx.)
So, runner A will take approximately 14.14 seconds to reach the meeting point.
Now, let's calculate how far runner B will have traveled in that time.
Distance traveled by runner B = speed * time
= 7 m/s * 14.14 s
= 98.98 meters (approx.)
Since the radius of the circular track is 200 meters (half the circumference), the distance from the starting point to the meeting point is half that distance, which is 100 meters.
Thus, runner A will reach the meeting point at 100 meters from the starting point and runner B will have traveled approximately 98.98 meters towards that point.
To find the angle at which they meet, we can use the formula:
Angle = (Distance / Circumference) * 360 degrees
Angle = (100 / 400) * 360
= 90 degrees
Therefore, runners A and B will meet at an angle of 90 degrees on the circular track.