Arunner is training for an upcoming marathon by running around a 80-m-diameter circular track at constant speed. let a coordinate system have its origin at the center of the circle with the x-axis pointing east and the y-axis north. the runner starts at (x, y) = (48 m, 0 m) and runs 2.5 times around the track in a clockwise direction.1. what is the magnitude of his displacement vector? 2. what is the direction of his displacement vector?

Explanation:

Given

diameter

runner start at (x,y)=(48 m,0 m)

Runner runs around the circle 2.5 times

i.e. he runs two and half rounds

displacement of two rounds is zero so net displacement is of half round

displacement

After 2.5 round he will be towards west direction with respect to origin

so direction of displacement vector is west

The gravitational force is  

Explanation:

The sketch describing this question is shown on the first uploaded image

from the sketch represent a unit vector pointing outward (This is shown the distance and the direction of the object  

Now the gravitational force of a object on the surface earth with mass m is

Mathematically represented as

Where is the mass o the earth and

          is the radius of the earth

The  minus sign shows that it is acting in the negative x-axis

Making the acceleration  due to gravity a subject of the formula we have

Considering when  the distance of the object to the center of the earth is r

Note: Any mass of the earth that is outside the distance covered by radius r as shown in the diagram does not affect the gravitational force

 The mass of the enclosed earth can be mathematically represented as

Where is the mass density  which is mathematically represented as

Now substituting these equation 3

Now substituting equation 5 into equation 1 to obtain the gravitational force as a function of distance r

From equation 2 we see that the

So substituting for g we have