(c) On Earth a fit astronaut without a space suit and life support pack can leap straight up and leave the ground with an initial vertical velocity of about v1y = 2.4 m/s and rise to a height of about h = 0.24 m above the point where he first loses contact with the ground. On the other hand, an astronaut on the moon is burdened with a space suit and life support pack to protect him from the rarified atmosphere and extreme temperatures. This gear weighs about 180 lbs on Earth. Under these conditions do you predict the Astronaut’s initial vertical velocity is less, the same, or more on the moon than it would be on the Earth without the gear? Explain the reasons for your answer.
To predict the astronaut's initial vertical velocity on the moon, we can use the concept of conservation of energy.
On Earth, when the astronaut leaps straight up without any external forces acting on them (such as air resistance), the only forces they need to overcome are their weight and the normal force from the ground. The work done by these forces can be calculated using the formula:
Work = Force * Distance * cos(theta)
In this case, since the astronaut is moving vertically upwards, the angle between the force and displacement vectors is 0 degrees, so cos(theta) = 1. Therefore, the work done is:
Work = Force * Distance
The work done by the astronaut's weight is negative because it acts in the opposite direction of displacement. The work done by the normal force is positive because it acts in the same direction as displacement. We can write this as:
Work = -mg * h + mg * h
where m is the mass of the astronaut and g is the acceleration due to gravity.
Since the astronaut is starting from rest and has no initial kinetic energy, the work done by these forces must equal the change in potential energy. Therefore, we have:
Potential Energy = -mg * h + mg * h
The negative sign indicates that the potential energy increases as the astronaut rises. Rearranging the equation, we find:
Potential Energy = mgh
Now, let's consider the case of the astronaut on the moon. On the moon, the gravitational acceleration is much smaller than on Earth, approximately 1/6th that of Earth's. Therefore, the force of gravity acting on the astronaut is reduced by a factor of 1/6. However, the weight of the gear remains the same.
Since the weight of the gear is an external force acting on the astronaut, it does not contribute to their potential energy. Therefore, the potential energy on the moon can be calculated as:
Potential Energy = mgh
Comparing the equation for potential energy on both Earth and the moon, we can see that they are identical. This means that the initial vertical velocity of the astronaut on the moon would be the same as on Earth without the gear.
In conclusion, the astronaut's initial vertical velocity on the moon would be the same as on Earth without the gear. The weight of the gear does not affect the potential energy and, therefore, does not change the initial vertical velocity.