Question

The asteroid Eros, one of the many minor planets that orbit the Sun in the region between Mars and Jupiter, has a radius of \(7.0 \mathrm{~km}\) and a mass of \(5.0 \times 10^{15} \mathrm{~kg} .(a)\) If you were standing on Eros, could you lift a \(2000-\mathrm{kg}\) pickup truck? \((b)\) Could you run fast enough to put yourself into orbit? Ignore effects due to the rotation of the asteroid. (Note: The Olympic records for the \(400-\mathrm{m}\) run correspond to speeds of \(9.1 \mathrm{~m} / \mathrm{s}\) for men and \(8.2 \mathrm{~m} / \mathrm{s}\) for women.)

Short answer

For part (a), assuming you can lift a certain weight on Earth, you would be able to lift a heavier weight on Eros due to the lower gravitational force. For part (b), the escape velocity on Eros is much higher than a human's typical running speed, so you cannot run fast enough to put yourself into orbit.

Step-by-step solution

Calculate the gravitational force exerted by Eros

We can use the equation for gravitational force \( F_g = G * m1 * m2/r^2 \) where \( G = 6.674 * 10^{-11} N(m/kg)^2 \), m1 is the mass of the pickup truck (2000 kg), m2 is the mass of Eros \( 5.0 * 10^{15} kg \), and r is the radius of Eros (7000 m). After calculating, the gravitational force will be smaller than on earth.

Determine if you can lift a pickup truck on Eros

On Earth, the force experienced by a 2000-kg object is \( F = m * g \) where g is acceleration due to gravity (9.8 m/s^2). If the force needed to lift the truck on Eros is less than the force experienced by the truck on Earth, then it could be lifted on Eros. So, find that force and make a comparison.

Calculate the escape velocity on Eros

The equation for escape velocity is \( v = \sqrt{2*G*M/r} \), where G is the gravitational constant, M is the mass of the celestial body, and r is the distance from the center of the celestial body (or radius for planets and asteroids). Calculate the escape velocity and compare it to a human's running speed.

Determine if you can put yourself into orbit

If the calculated escape velocity on Eros is lower than the average speed of an Olympic runner (9.1 m/s for men and 8.2 m/s for women), then it would be possible for a human to run fast enough to reach escape velocity and put themselves into orbit.

Key concepts

Gravitational Force

Gravitational force is an essential concept in understanding how objects interact with each other in the universe. It can be thought of as the invisible pull that all objects with mass exert on one another. The force of gravity depends on two main factors: the masses of the objects and the distance between them.

When considering the asteroid Eros, it's important to recognize that its gravitational pull is much weaker than Earth's. For example, if you're standing on Eros and trying to lift a 2000-kg pickup truck, you calculate the gravitational force it exerts using the formula \( F_g = \frac{G \cdot m_1 \cdot m_2}{r^2} \), where:

• \( G \) is the gravitational constant, \( 6.674 \times 10^{-11} \, \text{N}\cdot\text{m}^2/\text{kg}^2 \)
• \( m_1 \) is the mass of the truck \( 2000 \, \text{kg} \)
• \( m_2 \) is the mass of Eros \( 5.0 \times 10^{15} \, \text{kg} \)
• \( r \) is the radius of Eros \( 7000 \, \text{m} \)

Once calculated, you'd find that the force required to lift the truck on Eros is much less than what you'd need on Earth because the gravitational acceleration is considerably smaller.

Ultimately, understanding gravitational force helps explain why you could lift heavier things on celestial bodies like Eros, where gravity is weaker, compared to Earth.

Escape Velocity

Escape velocity is the speed an object must reach to break free from the gravitational pull of a celestial body, without any further propulsion. For instance, to escape Earth's gravity, a spacecraft needs to travel at around 11.2 km/s. The concept of escape velocity is crucial for scenarios like launching satellites or space exploration missions.

For the asteroid Eros, you can find the escape velocity by using the equation: \[ v = \sqrt{\frac{2 \cdot G \cdot M}{r}} \]where:

• \( v \) is the escape velocity
• \( G \) is the gravitational constant
• \( M \) is the mass of Eros \( 5.0 \times 10^{15} \, \text{kg} \)
• \( r \) is the radius \( 7000 \, \text{m} \)

When you calculate this for Eros, you find a much lower escape velocity than on Earth, due to its smaller mass and size.

Interestingly, this low escape velocity raises the hypothetical possibility that a human could theoretically "run" into orbit if they could reach the necessary speed, showcasing the unique aspects of physics on smaller celestial bodies.

Orbital Mechanics

Orbital mechanics is the study of the motions of objects in space, focusing on how the gravitational forces between celestial bodies affect their paths. This field is integral for understanding orbits of planets, asteroids, and artificial satellites.

When you consider placing yourself in orbit around Eros, you're diving into the realm of orbital mechanics. Achieving orbit involves reaching a specific balance where an object is moving forward enough to counteract the gravitational pull pulling it downwards.

For a human attempting to run into orbit on Eros, you'd first think about how this asteroid’s small mass and radius dictate a lower escape velocity. Using our previous calculations, if this escape velocity is within the range of human running speeds (8.2 m/s for women, 9.1 m/s for men), it suggests an intriguing scenario where solo orbital insertion might theoretically be achievable.

Though such a feat is purely theoretical, it highlights how orbital mechanics can offer fascinating insights into how celestial bodies interact and the possibilities they present for exploration.