A particle is projected from the surface of Earth with a speed equal to 2.4 times the escape speed. When it is very far from Earth, what is its speed?
?km/s
To find the speed of the particle when it is very far from Earth, we need to understand the concept of escape speed.
Escape speed is the minimum speed required for an object to escape the gravitational pull of a planet or celestial body. On Earth, the escape speed is given by the formula:
vesc = √(2GM/r)
Where:
- vesc is the escape speed
- G is the gravitational constant (approximately 6.674 × 10^-11 N(m/kg)^2)
- M is the mass of the planet/celestial body (in this case, Earth)
- r is the distance between the particle and the center of the planet (Earth's radius)
Now, if the particle is projected with a speed equal to 2.4 times the escape speed, we can calculate the actual escape speed (v_esc) by dividing the given speed by 2.4:
v_esc = given speed / 2.4
Then, when the particle is very far from Earth, it will have the same speed as the escape speed because it has already surpassed the escape speed. Therefore, the speed of the particle when it is very far from Earth would be equal to v_esc.
Now, using the known values, we can calculate the escape speed and the final speed:
1. Determine the escape speed:
v_esc = given speed / 2.4
2. Calculate the given speed:
Given speed = 2.4 times the escape speed
3. Substitute the given speed back into the formula to calculate v_esc:
v_esc = (2.4 × v_esc) / 2.4
4. Solve for v_esc:
v_esc = v_esc
Although it may seem that we are stuck with v_esc = v_esc, the values will cancel when solving the equation, which means that any value for v_esc would satisfy the equation. Therefore, we cannot determine the specific numerical value of v_esc, but we can conclude that the speed of the particle when it is very far from Earth will be equal to the escape speed.
To find the speed of the particle when it is very far from Earth, we need to understand the concept of escape speed. The escape speed is the minimum speed required for an object to completely escape the gravitational pull of a planet.
The escape speed, ve, can be calculated using the formula:
ve = sqrt(2 * G * M / R),
where G is the universal gravitational constant, M is the mass of the planet, and R is the distance from the center of the planet to the point from where the particle is projected.
Let's assume that the escape speed is v_e, and the speed at which the particle is projected is v_0 = 2.4 * v_e.
When the particle is very far from Earth, it means that it has escaped Earth's gravitational pull and is no longer affected by it. In this scenario, the only force acting on the particle is the force of gravity from other celestial bodies, which may be considered negligible.
Therefore, the speed of the particle when it is very far from Earth will remain constant and equal to the speed at which it was originally projected, which is 2.4 times the escape speed.
As we don't have the specific values for the escape speed, the mass of Earth, or the distance from the center of Earth to the point of projection, we can't provide an exact numerical value for the speed of the particle when it is very far from Earth. However, based on the given information, we know it is 2.4 times the escape speed.