Two soccer players, Mary and Jane, begin running from nearly the same point at the same time. Mary runs in an easterly direction at 4.43 m/s, while Jane takes off in a direction 60.2o north of east at 5.07 m/s. How long is it before they are 24.7 m apart?
What is the velocity of Jane relative to Mary? Enter first the x-component and then the y-component.
How far apart are they after 4.08 s?
Why did Mary and Jane start running? Did someone say there was free pizza at the finish line? Either way, let's solve this problem!
To find out how long it takes for Mary and Jane to be 24.7 m apart, we need to break down their velocities into their x and y components.
Mary is running purely in the east direction, so her x-component is 4.43 m/s, and her y-component is 0 m/s.
Jane is running in a direction 60.2° north of east, so we need to find the x and y components of her velocity. To do that, we can use trigonometry. The angle between east and Jane's direction is 60.2°, which means the angle between Jane's velocity and east is 90° - 60.2° = 29.8°.
The x-component of Jane's velocity is then 5.07 m/s × cos(29.8°), and the y-component is 5.07 m/s × sin(29.8°). Crunching the numbers gives us an x-component of 4.453 m/s and a y-component of 2.354 m/s.
Now, let's find the relative velocity of Jane with respect to Mary. This is as exciting as a soccer game! The x-component is the difference between Jane's and Mary's x-components, so it's 4.453 m/s - 4.43 m/s = 0.023 m/s. The y-component is the difference between their y-components, so it's 2.354 m/s - 0 m/s = 2.354 m/s.
So, the velocity of Jane relative to Mary is 0.023 m/s in the x-direction and 2.354 m/s in the y-direction.
Now, let's see how far apart they are after 4.08 s. We can use the formula distance = speed × time for both Mary and Jane. Mary's distance is 4.43 m/s × 4.08 s = 18.0464 m. Jane's distance is sqrt((4.453 m/s)^2 + (2.354 m/s)^2) × 4.08 s = 19.0607 m. Adding up their distances gives us a total distance of 18.0464 m + 19.0607 m = 37.1071 m.
So, after 4.08 s, Mary and Jane are about 37.1071 m apart. They're probably wondering, "Why are we running away from each other? We could've just shared the pizza!"
To find how long it takes for Mary and Jane to be 24.7 m apart, we can use the formula for relative velocity:
Relative velocity = √((v1x - v2x)^2 + (v1y - v2y)^2)
where v1x and v1y are the x and y components of Mary's velocity, and v2x and v2y are the x and y components of Jane's velocity.
Given:
v1x = 4.43 m/s (Mary's velocity in the x direction)
v2x = 5.07 m/s * cos(60.2o) (Jane's velocity in the x direction)
v1y = 0 m/s (Mary's velocity in the y direction)
v2y = 5.07 m/s * sin(60.2o) (Jane's velocity in the y direction)
Calculating:
v2x = 5.07 m/s * cos(60.2o) = 2.52 m/s
v2y = 5.07 m/s * sin(60.2o) = 4.38 m/s
Relative velocity = √((4.43 m/s - 2.52 m/s)^2 + (0 m/s - 4.38 m/s)^2)
Relative velocity = √(1.91^2 + (-4.38)^2)
Relative velocity = √(3.6481 + 19.1844)
Relative velocity ≈ √22.8325
Relative velocity ≈ 4.77 m/s
So, the velocity of Jane relative to Mary is approximately 4.77 m/s in the direction from Mary to Jane.
To find how far apart they are after 4.08 s, we can use the formula:
Distance = Relative velocity * time
Given:
Relative velocity = 4.77 m/s
time = 4.08 s
Calculating:
Distance = 4.77 m/s * 4.08 s
Distance ≈ 19.46 m
After 4.08 s, Mary and Jane are approximately 19.46 m apart.
To solve the first question, we can use vector addition to find the relative velocity of Jane with respect to Mary.
First, let's break down the velocities into their x and y components. Mary's velocity is entirely in the x-direction because she is running east, so her velocity vector can be written as (4.43 m/s, 0 m/s).
Jane's velocity makes a angle of 60.2 degrees north of east. To find the x and y components of her velocity, we can use trigonometry. The x-component can be found using the cosine function:
x-component of Jane's velocity = 5.07 m/s * cos(60.2 degrees)
The y-component can be found using the sine function:
y-component of Jane's velocity = 5.07 m/s * sin(60.2 degrees)
Now we have both components in the x and y directions.
To find the relative velocity of Jane with respect to Mary, we subtract Mary's velocity components from Jane's velocity components:
Relative velocity = (x-component of Jane's velocity - x-component of Mary's velocity, y-component of Jane's velocity - y-component of Mary's velocity)
Now, let's solve the equations to find the relative velocity components.
velocity of Jane relative to Mary = (5.07 m/s * cos(60.2 degrees) - 4.43 m/s, 5.07 m/s * sin(60.2 degrees) - 0 m/s)
Simplifying this:
velocity of Jane relative to Mary = (0.558 m/s, 2.601 m/s)
Therefore, the velocity of Jane relative to Mary is (0.558 m/s, 2.601 m/s).
Now, let's move on to the second question.
To find how far apart Mary and Jane are after 4.08 s, we can use the relative velocity we just calculated.
We know that distance = velocity x time.
So, the distance Mary and Jane are apart after 4.08 s is:
distance = magnitude of the relative velocity x time
distance = sqrt((0.558 m/s)^2 + (2.601 m/s)^2) * 4.08 s
distance = 2.725 m/s * 4.08 s
distance = 11.124 m
Therefore, after 4.08 s, Mary and Jane are 11.124 m apart.
Y = 5.07*sin60.2 = 4.40 m/s.
d = V*T = 24.7, 4.4*T = 24.7, T = 5.61 s.