An Earth satellite moves in a circular orbit 765 km above the Earth's surface. The period of the motion is 99.9 min.

(a) What is the speed of the satellite?

______ m/s

(b) What is the magnitude of the centripetal acceleration of the satellite?

______m/s^2

To find the speed of the satellite, we can use the formula:

v = (2πr) / T

where v is the speed, r is the radius of the orbit, and T is the period of the motion.

Given that the radius of the orbit is 765 km above the Earth's surface, we can convert it to meters:

r = 765 km = 765,000 m

The period of the motion is 99.9 min, which we need to convert to seconds:

T = 99.9 min = 99.9 * 60 s = 5994 s

Now, substitute the values into the formula to find the speed:

v = (2π * 765,000) / 5994

Calculating the expression:

v ≈ 4947 m/s

Therefore, the speed of the satellite is approximately 4947 m/s.

To find the magnitude of the centripetal acceleration, we can use the formula:

a_c = v^2 / r

Substitute the known values:

a_c = (4947)^2 / 765,000

Calculating the expression:

a_c ≈ 31.89 m/s^2

Therefore, the magnitude of the centripetal acceleration of the satellite is approximately 31.89 m/s^2.

To find the speed of the satellite, you can start by using the formula for the circumference of a circle:

C = 2πr

where C is the circumference and r is the radius of the circle. Since the satellite is moving in a circular orbit, the distance it travels in one complete revolution is equal to the circumference of the orbit.

(a)

Step 1: Convert the distance above Earth's surface from kilometers to meters:

765 km = 765,000 meters

Step 2: Calculate the radius of the orbit by subtracting the radius of the Earth from the distance above the Earth's surface:

Radius of orbit = distance above Earth's surface - radius of Earth

Radius of orbit = 765,000 m - 6,371,000 m

Radius of orbit = -5,606,000 m

Note: The negative sign is because the satellite orbits above the Earth's surface.

Step 3: Calculate the circumference of the orbit using the formula:

Circumference = 2π(radius of orbit)

Circumference = 2π(-5,606,000 m)

Circumference = -35,262,029.41 m

Note: The negative sign for the circumference is irrelevant, as speed is a scalar quantity and is always positive.

Step 4: Calculate the speed of the satellite using the formula:

Speed = Circumference / period

Speed = -35,262,029.41 m / 99.9 min

Speed = -353,103.32 m/min

Note: Again, the negative sign is irrelevant for speed, so we can convert minutes to seconds and take the absolute value of the speed to get the final answer.

To convert minutes to seconds, multiply by 60:

Speed = 353,103.32 m/min * (1 min / 60 s)

Speed ≈ 5885.06 m/s

(b)

The centripetal acceleration of an object moving in a circular orbit can be calculated using the formula:

ac = v^2 / r

where ac is the centripetal acceleration, v is the speed of the satellite, and r is the radius of the orbit.

Step 1: Use the absolute value of the speed calculated in part (a):

v = 5885.06 m/s

Step 2: Use the negative value of the radius of the orbit calculated in part (a):

r = -5,606,000 m

Step 3: Calculate the centripetal acceleration using the formula:

ac = (5885.06 m/s)^2 / -5,606,000 m

ac ≈ 6.20 m/s^2

So, the magnitude of the centripetal acceleration of the satellite is approximately 6.20 m/s^2.

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