How far from the surface of the earth does a rocket travel, before momentarily coming to rest, when it is launched with a initial velocity of 1/3 the scape speed? Uploa
Gravitational force
In nature, every object is attracted by every other object. This phenomenon is called gravity. The force associated with gravity is called gravitational force. The gravitational force is the weakest force that exists in nature. The gravitational force is always attractive.
Acceleration Due to Gravity
In fundamental physics, gravity or gravitational force is the universal attractive force acting between all the matters that exist or exhibit. It is the weakest known force. Therefore no internal changes in an object occurs due to this force. On the other hand, it has control over the trajectories of bodies in the solar system and in the universe due to its vast scope and universal action. The free fall of objects on Earth and the motions of celestial bodies, according to Newton, are both determined by the same force. It was Newton who put forward that the moon is held by a strong attractive force exerted by the Earth which makes it revolve in a straight line. He was sure that this force is similar to the downward force which Earth exerts on all the objects on it.
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![**Educational Content: Physics - Rocket Motion**
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### Problem Statement:
**How far from the surface of the Earth does a rocket travel, before momentarily coming to rest, when it is launched with an initial velocity of 1/3 the escape speed?**
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#### Explanation:
This problem involves calculating the maximum distance a rocket reaches from the Earth's surface when it is launched with an initial velocity that is a fraction of the escape velocity. The escape velocity is the speed at which an object must travel to completely overcome Earth's gravitational pull without further propulsion.
### Key Concepts:
1. **Escape Velocity:** The minimum velocity needed for an object to escape from the gravitational influence of a celestial body.
2. **Initial Velocity:** The speed at which the rocket is initially launched.
3. **Gravitational Potential Energy:** The energy an object possesses because of its position in a gravitational field.
### Approach and Solution:
To solve this problem, we can use the principle of conservation of energy. The total mechanical energy (kinetic + potential) of the system remains constant if only conservative forces (like gravity) are doing work.
#### Step-by-Step Solution:
1. **Determine Escape Velocity (v_e):** The escape velocity from the Earth is given by:
\[
v_e = \sqrt{\frac{2GM}{R}}
\]
where:
- \(G\) is the gravitational constant.
- \(M\) is the mass of Earth.
- \(R\) is the radius of Earth.
2. **Initial Kinetic Energy (KE_i) using fraction of Escape Velocity:**
The rocket is launched with an initial velocity \(v_i = \frac{1}{3}v_e\). Therefore, the initial kinetic energy is:
\[
KE_i = \frac{1}{2}mv_i^2 = \frac{1}{2}m\left(\frac{1}{3}v_e\right)^2 = \frac{1}{18}mv_e^2
\]
3. **Initial Potential Energy (U_i):**
The initial potential energy when the rocket is at the Earth's surface is:
\[
U_i = -\frac{GMm}{R}
\]
4. **Mechanical Energy at Maximum Height:**
At the maximum height, the rocket momentarily comes to rest, so its kinetic energy is zero. Let \(h\) be](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F38a5f9d6-9ba6-414e-a3d3-92c69a7c04d9%2F09423dd3-602e-4a9b-a53b-6fed68dca1e9%2F3bk46ym_processed.jpeg&w=3840&q=75)
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