Question

Ракете сообщили на полюсе Земли вертикальную скорость V=1.6 км/с. Зная радиус Земли R = 6400 км и ускорение свободного падения g=9.8 м/с2, найдите высоту h (в км), на которую поднимется ракета. Сопротивлением воздуха пренебречь. Подсказка: воспользоваться законом сохранения энергии. Ответ округлите до целого числа.

Answer 1

По закону сохранения энергии максимальная кинетическая энергия тела равна его максимальной потенциальной энергии т.е:

mv^2/2=mgh
сократим и выразим h:
h=v^2/2g=/(1600)^2/2*9.8=137634

Answer 2

Calculation of Rocket Height

To find the height to which the rocket will rise, we can use the law of conservation of energy. The total mechanical energy of the rocket is conserved throughout its motion. At the initial position, the rocket has kinetic energy due to its vertical velocity, and at the highest point of its trajectory, it has potential energy due to its height.

The formula for the conservation of energy is:

Initial kinetic energy + Initial potential energy = Final kinetic energy + Final potential energy

The initial kinetic energy is given by:

KE_initial = (1/2) * m * V^2

where: - m is the mass of the rocket (which we can assume to be constant) - V is the vertical velocity of the rocket (given as 1.6 km/s)

The initial potential energy is given by:

PE_initial = m * g * h_initial

where: - g is the acceleration due to gravity (given as 9.8 m/s^2) - h_initial is the initial height of the rocket (which we need to find)

At the highest point of the rocket's trajectory, the final kinetic energy is zero because the vertical velocity becomes zero. Therefore, the final potential energy is equal to the total mechanical energy of the rocket.

PE_final = m * g * h_final

Setting up the conservation of energy equation:

(1/2) * m * V^2 + m * g * h_initial = 0 + m * g * h_final

Simplifying the equation:

(1/2) * V^2 + g * h_initial = g * h_final

Now we can solve for the height h_final:

h_final = (1/2) * V^2 / g + h_initial

To find the height h_final, we need to know the initial height h_initial. In this case, the rocket is launched from the surface of the Earth, so the initial height is equal to the radius of the Earth (R = 6400 km).

Substituting the given values into the equation:

h_final = (1/2) * (1.6 km/s)^2 / (9.8 m/s^2) + 6400 km

Converting the units to match:

h_final = (1/2) * (1600 m/s)^2 / (9.8 m/s^2) + 6400 km

Calculating the height:

h_final = 1280000 m^2/s^2 / 9.8 m/s^2 + 6400 km

h_final = 130612.24 m + 6400 km

h_final = 130612.24 m + 6400000 m

h_final = 6530612.24 m

Rounding the answer to the nearest whole number:

h_final ≈ 6530612 km

Therefore, the rocket will rise to a height of approximately 6530612 km.