An artificial satellite is moving in a circular orbit with a speed of 6.5 km/s and a period of 80.0 min. A retarding rocket fires in a direction opposite to the motion. This provides a retarding deceleration of 21.0 m/s2. What is the acute angle between the radius vector and the total acceleration of the satellite?
you have two acceleation vectors.
One is gravity.
Two is the rocket firing
Both of those is at 90 degrees, so the angle can be found from the tangent when adding those.
Gravity: v^2/r
you are given v (convert to m/s), but r can be found from
v^2/r=g(re/r)^2
or r=g*(rearth/v)^2
gravity acceleration= v^2/g(re/v)^2
figure that out, it is pointed down. Check my math.
Radial acceleration: 21m/s^2
angle= arctan (radial acceleration/gravityacceleration)
check all this , I did it in my head.
bobpursley's answer is wrong. The question isn't a satellite revolving earth anyways.
Total Acceleration will consist of Centripetal (which is in the same direction as the r vector) and Tangential (which at any given moment will be your retarding one. With that we can set up a triangle that gives us the angle needed.
Since we don't know radius we can use V=r2Pi/T and solve for Centripetal.
We get V2Pi/T= Ac. From Ac=V^2/r and using our solve for r from above.
Our answer will be |arctan( Aretarding/Ac)|:
|arctan(21/(V2Pi/T))=67.96deg
To find the acute angle between the radius vector and the total acceleration of the satellite, we need to consider the different forces and motion involved.
First, let's break down the forces acting on the satellite:
1. Centripetal force: This force is responsible for keeping the satellite in a circular orbit. It acts towards the center of the orbit and is provided by the gravitational force between the satellite and the Earth.
2. Thrust force from the rocket: This force is opposite to the motion of the satellite and provides the retarding deceleration.
Next, let's calculate the centripetal force acting on the satellite:
The centripetal force is given by the equation: Fc = (m * v^2) / r, where Fc is the centripetal force, m is the mass of the satellite, v is the velocity of the satellite, and r is the radius of the orbit.
To find Fc, we need the mass of the satellite. However, since it's not provided in the question, we can assume it cancels out during the calculation of the angle.
Now, let's calculate the centripetal force (Fc) using the velocity and period given in the question:
Fc = (m * v^2) / r
We know that the speed of the satellite (v) is 6.5 km/s, which is equal to 6500 m/s since 1 km = 1000 m.
Given that the period (T) is equal to 80.0 minutes, we need to convert it to seconds:
T = 80.0 min * 60 s/min = 4800 s
The radius (r) of the orbit is not explicitly given in the question. However, we can calculate it using the period and the formula for the time period of a circular orbit:
T = 2πr / v
Rearranging the formula to solve for r:
r = (T * v) / (2π)
Substituting the values, we get:
r = (4800 s * 6500 m/s) / (2π)
Now, we have all the values to calculate the centripetal force (Fc):
Fc = (m * v^2) / r
Next, let's calculate the acceleration due to the retarding rocket (ar):
Given that the retarding deceleration (a) is 21.0 m/s^2, and the radius vector points towards the center of the circular orbit, we can say that ar = -a, where the negative sign denotes the opposite direction to the motion of the satellite.
Now, let's find the total acceleration (atot):
Since acceleration is a vector quantity, the total acceleration (atot) is the vector sum of the acceleration due to the retarding rocket (ar) and the centripetal acceleration (ac):
atot = ar + ac
Finally, let's find the acute angle (θ) between the radius vector and the total acceleration using the dot product:
θ = arccos((atot . r) / (|atot| * |r|))
Substituting the values we calculated, we can use this equation to find the angle.