Classical Mechanics Answers

A small stone of mass m, is thrown vertically upward with initial speed ,u. If the air resistance at speed v is mkv2, where k is a positive constant. Show that the stone returns to its starting point with speed given by v=u/((1+(ku2/g))1/2)

10. If a is a unit vector which is a function of a scalar prove that da/dt, in general a and da/dt are at right angles.

A small branch is wedged under a 200 kg rock and rests on a smaller object. The small object that creates a pivot point is called the fulcrum. The smaller object is 2.0 m from the large rock and the branch is 12.0 m long.
If the mass of the branch is negligible, what force must be exerted on the free end to just barely lift the rock?

What is normal modes of linear triatomic molecule, when is rotating around its center of mass with angular velocity omega?
The middle molecule mass is M
and the other molecules mass is m
and the spring constant is k
normal modes of linear triatomic molecule is soleved in Herbert Goldstein's classical mechanics book, when is not rotating.

A horizontal 781 N merry-go-round of radius 1.75 m is started from rest by a constant
horizontal force of 75 N applied tangentially to the merry-go-round.
Find the kinetic energy of the merry-goround after 3.97 s. The acceleration of gravity
is 9.8 m/s2
Assume the merry-go-round is a solid cylinder.
Answer in units of J.

You are an investigator at the scene of a car crash. A Volvo of has hit an Alpha Romeo from the side and the two are inter-tangled (stuck together). You can see that the Volvo was traveling due east on a road with a 50 mph speed limit and made a 30.0 m long skid mark before reaching the intersection where it hit the Alpha Romeo which was driving due North on a road with a 25 mph speed limit. There are additional skid marks from both cars that are 12.25 m long at 15 degrees North of East. Further investigation tells you that the mass of the Volvo (including the driver) is 1650 kg and the mass of the Alpha Romeo (including the driver) is 975 kg. You recognize that the accident occurred on dry concrete roads, each of which had a 45 mph speed limit. Your task is to determine if either car was speeding. 1. Determine the speed of the Volvo before it slammed on the brakes. 2. Find the speed of the Alpha Romeo when it was hit by the Volvo.

An arrow was shot at a target X=2.2m away. The speed of the arrow was 14.6m/s. Using g=9.8m s-2, and if the arrow was fired at a height of H=1.8m, what height above the ground (Y) does the bullseye need to ensure it is struck by the arrow (0.1m)?

See diagram below:
https://cdn.discordapp.com/attachments/523069000775761934/663875634237538336/Projectile_-_Horizontal_wall_1.PNG

An arrow was shot at a target X=2.2m away. The speed of the arrow was 14.6m/s. Using g=9.8m s-2, and if the arrow was fired at a height of H=1.8m, what height above the ground (Y) does the bullseye need to ensure it is struck by the arrow (0.1m)?

See diagram below:
https://cdn.discordapp.com/attachments/523069000775761934/663875634237538336/Projectile_-_Horizontal_wall_1.PNG

Bungee jumping: a man falls from the tower H = 200 m height with a rubber tope attached to his feet. The minimal height down to which the jumper will descend is h′ = 50 m over the ground. After all the oscillation will stop, the jumper will hang at the height h = 90 m over the ground. What maximal speed will the jumper reach during this jump? The air resistance may be neglected. g = 10 м/с2
Please, express result is m/s and write it down rounded
up to 3 significant digits