To give solution to the exercise we must use the concepts of Torque, Vector magnitude and vector direction of the forces.
For the given problem we have to
In this way the torque acting on the particle as a function of distance and time is,
The net torque acting on the particle is
PART B) The direction of the torque is given by,
Therefore the torque direction is 48.04° below the x axis.
Answers:
a)
b)
Explanation:
This situation is a good example of the projectile motion or parabolic motion, in which the travel of the snowball has two components: <u>x-component</u> and <u>y-component</u>. Being their main equations as follows for both snowballs:
<h3><u>Snowball 1:</u></h3><u>x-component: </u>
(1)
Where:
is the initial speed of snowball 1 (and snowball 2, as well)
is the angle for snowball 1
is the time since the snowball 1 is thrown until it hits the opponent
<u>y-component: </u>
(2)
Where:
is the initial height of the snowball 1 (assuming that both people are only on the x axis of the frame of reference, therefore the value of the position in the y-component is zero.)
is the final height of the snowball 1
is the acceleration due gravity (always directed downwards)
<u>x-component: </u>
(3)
Where:
is the angle for snowball 2
is the time since the snowball 2 is thrown until it hits the opponent
<u>y-component: </u>
(4)
Having this clear, let's begin with the answers:
<h2>a) Angle for snowball 2</h2>Firstly, we have to isolate from (2):
(5)
(6)
Substituting (6) in (1):
(7)
Rewritting (7) and knowing :
(8)
(9)
(10) This is the point at which snowball 1 hits and snowball 2 should hit, too.
With this in mind, we have to isolate from (4) and substitute it on (3):
(11)
(12)
Rewritting (12):
(13)
Finding :
(14)
(15) This is the second angle at which snowball 2 must be thrown. Note this angle is lower than the first angle
.
<h2>b) Time difference between both snowballs</h2>
Now we will find the value of and
from (6) and (11), respectively:
(16)
(17)
(18)
(19)
Since snowball 1 was thrown before snowball 2, we have:
(20)
Finding the time difference between both:
(21)
Finally: