00:01
Hi there, so for this problem for par a, we are told that at the earth's surface, a projectile is launched straight up at a speed that is given.
00:13
So we are given the initial speed.
00:16
Is this value 7 ,500 meters per second? so the question is, what is the maximum height in kilometers? so for that we use an equation from kinematics that states that the final speed square is equal to the initial speed square minus two times the acceleration due to gravity times the maximum height.
00:43
Now, we know that at the maximum height, the speed is momentarily equals to zero.
00:50
So now we just need to solve for the maximum height.
00:53
So that will be the initial speed squared divided by two times the acceleration due to gravity.
00:59
And now we just simply substitute the values.
01:12
Now, using our calculator, we obtain a value of.
01:28
So the value that we obtained from this is equal to 2 ,869 ,897 .96.
01:53
In meters.
01:54
Of course, the problem states that we need to answer this in kilometers.
01:58
So to do that, we use a factor of conversion.
02:01
We know that one kilometer corresponds to 1 ,000 meters.
02:12
So we will have that the maximum height is approximately a value of 2 ,869 .9 .9.
02:27
Kilometers.
02:29
So that's a solution for the first part of this problem.
02:35
Now for the second part of this problem, we are asked about, well, we are told that a 1 ,000 kilograms satellite is in a circular orbit around the earth.
02:48
Now the question is how much energy is needed to boost the satellite from a radius, an initial radius, that is equal to two, hundred kilometers to a radius, a final radius of 300 kilometers.
03:13
So to solve this, we use the following equation.
03:18
That is that the change in the potential energy is equal to the gravitational constant, the mass of the earth times the mass of the satellite that we are given.
03:35
The mass is in this case 1 ,000 kilograms.
03:41
And this times 1 divided by the radius, the initial radius, minus 1 divided by the final radius.
03:52
And now we just need to simply substitute all of these values.
03:56
So the gravitational constant is equal to 6 .600 and 72 times 10 to the minus 11.
04:14
The mass of the earth is also given, and that is 5 .98 times 10 to the 24, and this times the mass of the satellite, which is 1 ,000.
04:32
And now we just simply substitute the values of the radius.
04:38
So remember that the value that we are given, you need to add the 200 kilometers or the 300 kilometers.
04:52
So for example, in this case, what we are going to have is, well, let's call this prime prime.
04:59
So the radius prime is this value that we have in here, plus, the radius of the earth.
05:07
Of course, that radius should be in meters, so this in here should be 200 ,000 kilometers, and this should be 300 ,000 kilometers...