A 1000 kg rocket without an internal propulsion system is launched with an initial speed of 7.00 x 102 m/s. Determine the maximum height attained by the rocket.

1/2 m v^2 = m g h

h = v^2 / (2 g)

To determine the maximum height attained by the rocket, we need to consider the rocket's initial speed and the gravitational force acting on it.

Step 1: Find the total energy of the rocket at the initial speed.
The total energy (E) of the rocket can be calculated using the formula:
E = (1/2)mv^2
where m is the mass of the rocket and v is its initial speed.

Given:
Mass of the rocket (m) = 1000 kg
Initial speed (v) = 7.00 x 10^2 m/s

Plugging in the values:
E = (1/2) * 1000 kg * (7.00 x 10^2 m/s)^2

Step 2: Find the potential energy of the rocket at its maximum height.
At the maximum height, the velocity of the rocket will be zero, and all its initial kinetic energy will be converted into potential energy.

Potential energy (PE) can be calculated using the formula:
PE = mgh
where m is the mass of the rocket, g is the acceleration due to gravity, and h is the maximum height.

Given:
Mass of the rocket (m) = 1000 kg
Acceleration due to gravity (g) = 9.8 m/s^2 (approximate)

Plugging in the values:
PE = 1000 kg * 9.8 m/s^2 * h

Step 3: Equate the total energy and potential energy to find the maximum height.
Since the total energy of the rocket is conserved, we can equate the initial kinetic energy (E) to the potential energy at maximum height (PE):

(1/2) * 1000 kg * (7.00 x 10^2 m/s)^2 = 1000 kg * 9.8 m/s^2 * h

Simplifying the equation:
3.5 x 10^8 kg m^2/s^2 = 9800 kg m^2/s^2 * h

Dividing both sides by 9800 kg m^2/s^2:
h = (3.5 x 10^8 kg m^2/s^2) / (9800 kg m^2/s^2)

Calculating the result:
h ≈ 3.57 x 10^4 m

Therefore, the maximum height attained by the rocket is approximately 3.57 x 10^4 meters.

To determine the maximum height attained by the rocket, we need to analyze the rocket's motion using the principles of projectile motion. We can break down the motion into vertical and horizontal components.

Given:
Mass of the rocket, m = 1000 kg
Initial speed of the rocket, v₀ = 7.00 x 10² m/s

1. Analyzing the vertical motion:
The only force acting on the rocket in the vertical direction is gravity, which will decelerate its upward motion until it reaches its maximum height and reverses direction. At the maximum height, the rocket momentarily comes to rest before falling back down.

Using the kinematic equation for vertical motion:
v² = u² + 2as

Where:
v = final velocity (which is zero when the rocket reaches the maximum height)
u = initial velocity (v₀)
a = acceleration (which is equal to the acceleration due to gravity, g)
s = displacement (which is equal to the maximum height, h)

Rearranging the equation, we get:
h = (v² - u²) / (2a)

Substituting the values:
u = 7.00 x 10² m/s
v = 0 m/s
a = acceleration due to gravity = -9.8 m/s² (negative since it acts in the opposite direction to the motion)

Calculating:
h = (0 - (7.00 x 10²)²) / (2 * -9.8)

2. Solving the equation:
h = (0 - 490000) / -19.6
h ≈ 25000 meters

Therefore, the maximum height attained by the rocket is approximately 25000 meters.