a particle is fired vertically upward from earth’s surface and it goes up to a height equal to earth’s radius . Find the initial speed of the particle

To determine the initial speed of a particle fired vertically upward from Earth's surface that reaches a height equal to Earth's radius, we can use the principles of gravitational potential energy and kinetic energy. This scenario involves the gravitational force acting on the particle as it ascends. Let's break down the problem step by step.

Understanding the Forces at Play

When the particle is launched, it has an initial kinetic energy due to its speed. As it rises, this kinetic energy is converted into gravitational potential energy until it reaches its maximum height, which is equal to the radius of the Earth (denoted as R).

Key Concepts

• Kinetic Energy (KE): The energy of the particle due to its motion, given by the formula KE = (1/2)mv², where m is the mass of the particle and v is its velocity.
• Gravitational Potential Energy (PE): The energy stored due to its position in a gravitational field, calculated as PE = -GMm/r, where G is the gravitational constant, M is the mass of the Earth, m is the mass of the particle, and r is the distance from the center of the Earth.

Setting Up the Energy Equation

At the surface of the Earth, the particle has an initial kinetic energy and zero potential energy (taking the potential energy at the surface as our reference point). At the maximum height (2R, where R is the radius of the Earth), the particle has zero kinetic energy (as it momentarily stops) and maximum potential energy.

The conservation of mechanical energy states that the total energy at the start equals the total energy at the maximum height:

Initial KE + Initial PE = Final KE + Final PE

Applying the Conservation of Energy

At the Earth's surface (initial state):

• Initial KE = (1/2)mv²
• Initial PE = 0

At the maximum height (final state):

• Final KE = 0
• Final PE = -GMm/(2R)

Now we can set up the equation:

(1/2)mv² + 0 = 0 - GMm/(2R)

Simplifying the Equation

We can simplify this equation by canceling out the mass (m) of the particle, assuming it is not zero:

(1/2)v² = GM/(2R)

Now, multiply both sides by 2:

v² = GM/R

Finally, take the square root to find the initial speed (v):

v = √(GM/R)

Calculating the Values

To find the numerical value of the initial speed, we need the values for G and M:

• Gravitational constant, G ≈ 6.674 × 10⁻¹¹ N(m/kg)²
• Mass of the Earth, M ≈ 5.972 × 10²⁴ kg
• Radius of the Earth, R ≈ 6.371 × 10⁶ m

Now, substituting these values into the equation:

v = √[(6.674 × 10⁻¹¹ N(m/kg)²)(5.972 × 10²⁴ kg) / (6.371 × 10⁶ m)]

Calculating this gives:

v ≈ √(9.81 × 10²) ≈ 31.3 m/s

Final Thoughts

The initial speed required for the particle to reach a height equal to Earth's radius is approximately 31.3 m/s. This example illustrates the interplay between kinetic and potential energy in a gravitational field, showcasing how energy conservation principles can be applied to solve real-world physics problems.