Your spaceship lands on an unknown planet. to determine the characteristics of this planet, you drop a 1.50 kg wrench from 5.50 m above the ground and measure that it hits the ground 0.811 s later. you also do enough surveying to determine that the circumference of the planet is 6.28×104 km . part a what is the mass of the planet, in kilograms?

The mass of the planet is

Explanation:

We have a first part of the problem, which we resolve with kinematics, knowing that

where d is given (5.50m), d₀ is zero, v₀ is zero too (as the wrench starts falling from static position), t is given (0.811s), and a is what we want to know for the second part of the problem. We clear a

Then for the second part, we use Newton's gravitational Law, where

m is the wrench mass, M is the planet mass, G is the gravity universal constant, and r is calculated from the given circumference (with the correct units) as

Finally, we have that

Therefore, replacing with the data calculated, and the known value of G, we can calculate M of the planet

russian is not my first language, so i'm afraid i can't write out an answer in russian, but hopefully you can find a way to do so. here's my best translation of the problem: you're given a plot of a particle's position in one-dimensional space over time, and you're asked to find either the total distance traveled by the particle after 6 seconds had elapsed, or the total displacement of the particle after 6 seconds.

if you want the total distance traveled, you would first need to determine the particle's velocity as a function of time, then integrate that over the 6-second interval. we know the position function is quadratic with roots at 0 and 6, so we have

where is unknown. because the position is parabolic, there is symmetry about the midpoint at , and we know that

so that the position function is

which means the velocity is

then the total distance traveled is

so the total distance traveled is 20 m.

on the other hand, if you want to find displacement: the particle starts at and ends at , so its displacement is 0.