Question Video: Finding the Radius of a Planet given Its Mass and the Time Taken by an Object at a Given Distance to Reach Its Surface

An astronaut dropped an object from a height of 2352 cm above the surface of a planet, and it reached the surface after 8 s. The mass of the planet is 7.164 × 10²⁴ kg, while that of the Earth is 5.97 × 10²⁴ kg, and the radius of the Earth is 6.34 × 10⁶ m. Given that the gravitational acceleration of the Earth is 𝑔 = 9.8 m/s², find the radius of the other planet.

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Video Transcript

An astronaut dropped an object from a height of 2352 centimeters above the surface of a planet, and it reached the surface after eight seconds. The mass of the planet is 7.164 times 10 to the power of 24 kilograms while that of the Earth is 5.97 times 10 to the power of 24 kilograms, and the radius of the Earth is 6.34 times 10 to the power of 6 meters. Given that the gravitational acceleration of the Earth is 9.8 meters per second squared, find the radius of the other planet.

So this problem is going to be solved in two parts. The first part of this problem is taking a look at a bit of information we’re given about an astronaut dropping an object from a height of 2352 centimeters above the surface of a planet and it reaching the surface after eight seconds because what we can do here is use one of our equations of constant acceleration to help us find what the acceleration due to gravity would be on that planet. And our equations of constant acceleration are sometimes known as SUVAT equations. And that’s because the letters we use to represent the variables. So we got 𝑠 which is our distance, 𝑢 is our initial velocity, 𝑣 is our final velocity, 𝑎 is our acceleration, and 𝑡 is our time.

Well, first of all, we have our distance which is 2352 centimeters. However, what we need to do is work in meters. So therefore, what we’re going to do is actually divide 2352 by 100 because there’re 100 centimeters in a meter. So therefore, this becomes 23.52 meters. Okay, next, we’re gonna move on to the initial velocity. Well, this is going to be zero meters per second because we’re told that the astronaut drops an object. So therefore, it’s starting from rest. Our final velocity we don’t know, and it’s not something we’re trying to find, so we can ignore that. Our acceleration due to gravity is what we’re trying to find. And what we’re gonna represent this as is 𝑔 sub 𝑝, so the gravity on the planet. And then finally, we know that the time is equal to eight seconds.

Okay, great, we have each of our variables, but which one of the equations of constant acceleration are we going to use to solve this part of the problem? Well, the equation we’re going to use is 𝑠 is equal to 𝑢𝑡 plus a half 𝑎𝑡 squared. And that’s because it contains each of the variables we’ve got or we’re actually looking for. So now, what we need to do is substitute in our values. So when we do that, we get 23.52 is equal to zero multiplied by eight plus a half multiplied by 𝑔 sub 𝑝 multiplied by eight squared. So therefore, we’re gonna get 23.52 equals 32𝑔 sub 𝑝. So now what we need to do is divide through by 32. And when we do that, we get a value of 𝑔 sub 𝑝 of 0.735. So this now tells us the acceleration due to gravity on the planet is 0.735 meters per second squared.

So now if we take a look at the information we’re given in the second part the question, we’ve got the mass of the planet, the mass of the Earth, the radius of the Earth, the acceleration due to gravity on the Earth, and also we’ve just worked that out on the planet, and we’re trying to find the radius of the other planet. So therefore, what we’re looking is at Newton’s law of universal gravitation. So because we’re looking at this, what we have is a formula to help us solve the problem. And that formula is 𝑔 is equal to capital 𝐺 multiplied by 𝑚 over 𝑟 squared. And this is where 𝑔 is the acceleration due to gravity, capital 𝐺 is the universal gravitational constant, 𝑚 is the mass, and 𝑟 is the radius.

So if we notice straightaway the bit of information that we haven’t been given in this problem is the universal gravitational constant. But in fact, what we can do is work that using the information we’ve been given. And we can do that using the information we’ve been given about the Earth. We’re told that the mass of the Earth is 5.97 times 10 to the power of 24 kilograms, the radius of the Earth is 6.34 times 10 to power of six meters, and the acceleration due to gravity on the Earth is 9.8 meters per second squared. So all we need to do is substitute this into our formula and rearrange to find capital 𝐺, the universal gravitational constant. So when we do this, we’re gonna get 9.8 equals capital 𝐺 multiplied by 5.97 times 10 to the power of 24 over 6.34 times 10 to power of six squared.

So now if we rearrange this by multiplying each side of the equation by 6.34 times 10 to power of six squared and dividing by 5.97 times 10 to power of 24, we have 𝐺 is equal to 9.8 multiplied by 6.34 times 10 to power of six squared over 5.97 times 10 to the power of 24. And this is gonna give us capital 𝐺 is equal to 6.598 continued multiplied by 10 to power of negative 11 meters cubed per kilogram per second squared. So we found that. So now what we can do is substitute the information we have in to find the radius of the other planet. And to enable us to do that, we’re going to clear some space on the left-hand side. The information that we are keeping is the fact that the acceleration due to gravity on the planet is equal to 0.735. And what we also know is that the mass of the planet is 7.164 times 10 to power of 24.

The radius of the planet is what we’re trying to find. And the universal gravitational constant we’ve worked out in this scenario to be 6.598 continued multiplied by 10 to power of negative 11. What I’ve done, in fact, is stored the rest of that value into the calculator so we get a really accurate result at the end. So therefore, what we get is 0.735 equals 6.598 multiplied by 10 to power of negative 11 multiplied by 7.164 multiplied by 10 to power of 24 over 𝑟 sub 𝑝 squared. And then what we need to do is multiply through by 𝑟 sub 𝑝 squared and divide by 0.735 which is gonna give us 𝑟 sub 𝑝 squared is equal to 6.431296 times 10 to power of 14. Then if we take the square root of both sides, we’re gonna get 𝑟 sub 𝑝 equals 25360000. So therefore, we can say that the radius of the other planet is going to be 2.536 times 10 to the power of seven meters.

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