Problem 45 - College Physics

Question

The space shuttle. During the first stage of its launch, a space shuttle goes from rest to 4973 \(\mathrm{km} / \mathrm{h}\) while rising a vertical distance of 45 \(\mathrm{km}\) . Assume constant acceleration and no variation in \(g\) over this distance. (a) What is the acceleration of the shuttle? (b) If a 55.0 \(\mathrm{kg}\) astronaut is standing on a scale inside the shuttle during this launch, how hard will the scale push on her? Start with a free-body diagram of the astronaut. (c) If this astronaut did not realize that the shuttle had left the launch pad, what would she think were her weight and mass?

Step-by-Step Solution

Answer

a) 21.2 m/s², b) 1705 N, c) 1705 N weight, 55.0 kg mass.

Step 1: Convert Velocity Units

First, we need to convert the shuttle's final velocity from kilometers per hour to meters per second. The conversion factor is 1 km/h = 0.27778 m/s. Thus, 4973 km/h is equivalent to \( 4973 \times 0.27778 \approx 1381.9 \) meters per second.

Step 2: Use Kinematic Equation to Find Acceleration

Using the kinematic equation \( v^2 = u^2 + 2as \), where \( v = 1381.9 \) m/s, \( u = 0 \), and \( s = 45000 \) m, we can solve for acceleration \( a \). Rearranging gives \( a = \frac{v^2}{2s} \). Plug in the values: \[ a = \frac{(1381.9)^2}{2 \times 45000} = 21.2 \, \text{m/s}^2 \].

Step 3: Calculate the Scale Force on the Astronaut

The net force \( F \) on the astronaut is the sum of gravitational force and the force exerted by the scale. Since \( F = ma \) for the total force exerted by the acceleration and gravity, where \( m = 55.0 \) kg, the apparent force due to acceleration is \( m \cdot a = 1166 \) N. The gravitational force \( mg = 539 \) N. The net force on the scale is therefore \( 1166 + 539 = 1705 \) N.

Step 4: Calculate the Apparent Weight and Realization

The apparent weight of the astronaut is equal to the reading of the scale, which is 1705 N. The apparent mass won't change due to acceleration and will remain at 55.0 kg. The astronaut, feeling the force as weight, would think her weight is equal to this scale reading and perceive her mass as unchanged.

Key Concepts

Constant Acceleration

When we talk about constant acceleration, we're referring to a scenario where the speed of an object increases steadily over time. This is often the case with a space shuttle during launch. Constant acceleration means the velocity of the shuttle is changing at a uniform rate.
To visualize this, imagine a car steadily picking up speed every second without any jerky movements. The same applies to the shuttle.
The main reason for assuming constant acceleration in physics problems is to simplify calculations and to use the kinematic equations that offer straightforward solutions. This doesn't mean the forces or conditions remain unchanged; it simply makes the math easier to handle.

Kinematic Equations

Kinematic equations are powerful tools used for calculating motion parameters when acceleration is constant. In the context of the space shuttle, these equations help in determining the shuttle's acceleration based on its change in velocity and distance traveled.
One of the key equations is:

• \( v^2 = u^2 + 2as \)

Here:

• \( v \) is the final velocity.
• \( u \) is the initial velocity (which is 0 for our shuttle).
• \( a \) is the acceleration.
• \( s \) is the distance traveled.

By rearranging this equation, we can solve for the acceleration \( a \), providing insight into how quickly the shuttle is picking up speed. This demonstrates how effective equations can be in unraveling motion under specific conditions.

Apparent Weight

Apparent weight refers to the net force that a person or object feels due to gravity and any other forces acting on them, like acceleration. In the case of the astronaut standing on a scale during shuttle launch, her apparent weight becomes the sum of the gravitational force and the force due to acceleration.
Apparent weight is crucial because it determines what the astronaut feels or perceives as her weight during the launch. If a scale reads higher than normal, it's because, in addition to gravity pulling her down, the shuttle's acceleration is pushing her upwards too. This combined force results in a higher apparent weight than when on the ground.

Newton's Laws of Motion

Newton's Laws of Motion form the foundation for understanding motion in physics. When analyzing the launch of a space shuttle, all three laws come into play:

• **First Law:** An object at rest remains at rest, and an object in motion continues in motion unless acted upon by an external force. This tells us why the shuttle doesn't move until massive engines provide a force.
• **Second Law:** This is expressed as \( F = ma \), where \( F \) is the net force applied, \( m \) is mass, and \( a \) is acceleration. It's essential for calculating how the shuttle's engines accelerate it from 0 to 4973 km/h.
• **Third Law:** For every action, there is an equal and opposite reaction. This explains why the shuttle lifts off; the exhaust gases push downwards, and the shuttle moves upwards.

Combining these laws allows us to fully understand the complex forces at play during a launch and how they affect everything on board.