Question

A space traveler whose mass is \(75.0 \mathrm{~kg}\) leaves Earth. Compute his weight \((a)\) on Earth, \((b)\) on Mars, where \(g=3.72\) \(\mathrm{m} / \mathrm{s}^{2}\), and \((c)\) in interplanetary space. \((d)\) What is his mass at each of these locations?

Short answer

The weight of the astronaut is 735 N on Earth, 279 N on Mars, and 0 N in interplanetary space. His mass is 75.0 kg at each of these locations.

Step-by-step solution

Weight on Earth

To find the weight on Earth, we use the given mass (75.0 kg) and the acceleration due to gravity on Earth, which is approximately 9.8 m/s². Thus, we calculate \( F_e = m \cdot g_e = 75.0 kg \cdot 9.8 m/s² = 735 N \). So, the weight of the astronaut on Earth is 735 N.

Weight on Mars

To find the weight on Mars, we use the mass of the astronaut (75.0 kg) and the provided acceleration due to gravity on Mars (3.72 m/s²). Thus, we calculate \( F_m = m \cdot g_m = 75.0 kg \cdot 3.72 m/s² = 279 N \). So, the weight of the astronaut on Mars is 279 N.

Weight in interplanetary space

In interplanetary space, there is effectively no gravitational force acting on the astronaut (or it's negligible). Thus, his weight in this environment would be 0 N, because weight is the force exerted by gravity on an object.

Mass at each location

The astronaut's mass remains constant at each location because mass is a measure of the amount of matter in an object. Thus, his mass is 75.0 kg on Earth, Mars, and in interplanetary space.

Key concepts

Weight Calculation

Understanding how to calculate weight is essential in physics. Weight is defined as the force exerted by gravity on an object. To compute this force, we use the formula: \[ F = m \cdot g \] where \( F \) is the weight in newtons (N), \( m \) is the mass in kilograms (kg), and \( g \) denotes the acceleration due to gravity in meters per second squared (m/s²).For example, an astronaut with a mass of \( 75.0 \mathrm{kg} \) on Earth, where \( g \) is approximately \( 9.8 \mathrm{m/s}^2 \), has a weight calculated as:\[ F_{\text{earth}} = 75.0 \mathrm{kg} \cdot 9.8 \mathrm{m/s}^2 = 735 \mathrm{N} \]. This simple formula allows you to understand how weight can change across different celestial bodies, like planets, due to the variation in \( g \).

Acceleration Due to Gravity

The acceleration due to gravity, symbolized by \( g \), varies depending on where in the universe you are. On Earth, it averages \( 9.8 \mathrm{m/s}^2 \), but on Mars, for example, \( g \) is \( 3.72 \mathrm{m/s}^2 \). The variation happens because \( g \) depends on the mass of the celestial body and the distance from its center to the object. More massive planets or those with denser cores exert a stronger gravitational pull resulting in higher values of \( g \).

To calculate the weight of an object on different planets, you simply need to substitute the appropriate value of \( g \) for that celestial body in the weight calculation formula. For the astronaut on Mars, the weight becomes:\[ F_{\text{mars}} = 75.0 \mathrm{kg} \cdot 3.72 \mathrm{m/s}^2 = 279 \mathrm{N} \].

Interplanetary Physics

Once an object, like an astronaut, leaves a planet and enters interplanetary space, the gravitational forces from nearby celestial bodies become considerably weaker. In the expanse of space between planets, the value of \( g \) approaches zero. This almost null gravity environment is what we experience as 'weightlessness' in space.

In such conditions, the weight of an object would technically be zero, because weight depends on the gravitational force, which is insignificant in interplanetary space. Therefore, the astronaut's weight is \( 0 \mathrm{N} \) when not under the influence of a celestial body's gravity.

Mass and Weight Difference

It's important not to confuse mass with weight. Mass is an intrinsic property of an object and remains constant regardless of location. It measures the amount of matter in an object, and is expressed in kilograms (kg). Weight, on the other hand, is the gravitational force acting on that mass and is measured in newtons (N).For example, an astronaut has a mass of \( 75.0 \mathrm{kg} \) which is the same whether they're on Earth, Mars, or floating in interplanetary space. However, the weight changes depending on the gravitational pull of the body they are on. As illustrated in the previous sections, the astronaut's weight is \( 735 \mathrm{N} \) on Earth and \( 279 \mathrm{N} \) on Mars, but would be essentially non-existent when in the vacuum of space between planets.