An object with a mass of 2kg falls without initial velocity and is subjected to air resistance which is directly proportional to the square of velocity (the coefficient of proportionality is k), Find the velocity expression of the object at any time T. (please use the differential equation solution)

f=kv^2
a=g-f/m=g-kv^2/m
dv/dt=g-kv^2/m
1/(g-kv^2/m) dv=dt
The expressions of T and V can be obtained by dividing the area into two parts
Using the integral formula: ∫ 1 / (x ^ 2-A ^ 2) DX = (1 / 2a) ln [(x-a) / (x + a)] + C

RELATED INFORMATIONS

1. On the afternoon of May 31, this year, there was a hailstorm in Beijing urban area. Assuming that the mass of M hail fell from the height of H, in the process of falling, the resistance was proportional to the square of the falling speed of the hail, and the proportional coefficient was K, then the maximum speed that the hail could reach was______ (it can be considered that the mass of hail will not change in the process of falling)
2. If the air resistance is directly proportional to the square of the object's velocity, and the air resistance is also directly proportional to the square of the object's cross-sectional area, can it be written as follows: If the air resistance is proportional to the square of the velocity of the object, the air resistance is also proportional to the square of the cross-sectional area of the object, Can it be written as: F = k.v & # 178;. S & # 178;
3. When a drop of rain falls from the air with enough height, the air resistance is proportional to the square of the velocity. How does the gravitational potential energy and kinetic energy of the drop change?
4. If two raindrops fall from the air, their masses are M1 and M2 respectively, Before landing on the ground, the power ratio of gravity is 0
5. If the resistance of an object with a mass of 2kg falling in the air is proportional to its velocity, the proportional coefficient k = 0.5N · s \, Try to find: 1, what is the final velocity of the object in the air? 2, when the velocity of the object is 20 m / s, what is the acceleration
6. If the air resistance is proportional to the square of the velocity, the proportional coefficient is k, and the mass of the satellite is m, then the velocity of the satellite is? The speed is represented by K and m
7. When the mass m hail falls from the height h, the resistance is directly proportional to the square of falling velocity, and the proportional coefficient is K. what is the maximum falling velocity of hail?
8. A small ball starts to fall from a standstill at a height of 18m. Suppose that the air resistance in the process of falling is always 0.1 times of gravity Find: (1) the acceleration of the ball (2) The speed of the ball when it lands
9. When an object with a weight of Mg falls vertically in the air, it is not only affected by gravity, but also by a resistance f which is proportional to the square of the velocity. Its size f = kV ^ 2, K is a positive constant. Then the final velocity of the falling object (that is, the velocity of the object when it finally moves at a constant speed) is______ ．
10. In the case of no wind, the raindrop falls vertically along a straight line in the air. The air resistance f is directly proportional to the square of the raindrop velocity, f = Kv2. If the mass of a raindrop is m, the maximum kinetic energy of the raindrop is
11. Suppose that the air resistance of parachute is directly proportional to the square of the falling speed, that is, f = kV ＾ 2, and the proportional coefficient k = 20n · s ＾ 2 / m ＾ 2 1) What do skydivers do in the air, 2) When the velocity reaches 4m / s, what is the falling acceleration? Is the first stage uniform acceleration? If so, why? Is it possible that the resistance is greater than gravity?
12. Let an object with mass m fall from rest in the air. The air resistance is proportional to the square of the falling speed of the object, and the proportional coefficient k > 0 Find the relationship between the falling speed of an object and time
13. A small ball is thrown vertically at the initial velocity of 21 M / s. suppose that the air resistance of the ball is 0.05 times of the gravity, then the maximum height of the ball is______ m. The speed of falling back to the throw point is_____ M / S (g = 10) m/s2).
14. When an object moves in the air along the vertical direction, and only under the action of gravity and air resistance, the motion of the object will be stable（ When an object moves in the air along the vertical direction, and only under the action of gravity and air resistance, the () a potential energy, B kinetic energy, C mechanical energy and d mechanical energy of the object may not be changed
15. When a charged ball moves from point m to point n in the air, it is respectively affected by gravity, air resistance and electric field force. In the process of movement, the electric field force does 14J work on the ball, the ball overcomes the air resistance and does 4J work, and the kinetic energy of the ball increases 8j A. The gravitational potential energy of the ball at M is more than that at N 2 & nbsp; JB. The mechanical energy of the ball at M is more than that at n 4 & nbsp; JC. The mechanical energy of the ball at M is 10 & nbsp; more than that at n JD. The electric potential energy of the ball at M is 14 & nbsp; more than that at n J
16. If the ratio of the initial velocity to the velocity back to the throwing point is k, and the air resistance is constant during the motion, the ratio of gravity to resistance is K______ ．
17. A basketball with a gravity of 5N is thrown vertically. It is assumed that the resistance of air to the basketball is 0.5N, Then the resultant forces of the basketball in the process of vertical rising and falling are (10N / kg) A.5.5N,5N B.5.5N,4.5N C.4.5N,5.5N D.5N,4.5N
18. The ball with mass of M = 1kg rolls down from point a at the upper end of 1 / 4 smooth circular arc groove with radius r of 1, and leaves the circular arc horizontally at point B at the height h = 3M above the ground (1) The velocity VA when leaving the circular groove (2) Landing speed VC size (3) The work done by gravity throughout the process
19. At a height of 45m from the ground, a small ball with a mass of 0.1kg starts to fall freely. The instantaneous power of gravity at the end of the first second? The average power of gravity work in the second second second?
20. There is a steel plate placed horizontally on the ground. There is a steel ball with mass m = 1kg at 3M above it. It moves vertically downward at the initial velocity V0 = 2m / s. assuming that the ball is subject to a constant air resistance f = 2n, there is no kinetic energy loss when the ball collides with the steel plate, and the ball finally stops moving (1) What's the speed of the first bounce? (2) The height of the first bounce? (3) What's the distance s it takes to stop?