Problem 74 - University Physics with Modern Physics

Question

An elevator is moving upward at a constant speed of 2.50 m/s. A bolt in the elevator ceiling 3.00 m above the elevator floor works loose and falls. (a) How long does it take for the bolt to fall to the elevator floor? What is the speed of the bolt just as it hits the elevator floor (b) according to an observer in the elevator? (c) According to an observer standing on one of the floor landings of the building? (d) According to the observer in part (c), what distance did the bolt travel between the ceiling and the floor of the elevator?

Step-by-Step Solution

Answer

(a) 0.782 sec; (b) 7.67 m/s; (c) 5.17 m/s; (d) 1.96 m.

Step 1: Analyze the Problem (a)

We need to determine the time it takes for the bolt to hit the floor of the elevator. Since the bolt is falling in the elevator, we'll analyze it from the elevator's perspective. The elevator's own motion does not influence the time of fall as gravity will act the same inside the elevator as outside. The initial height is 3.00 meters, and the initial velocity (relative to the elevator) is 0 m/s. Use the equation for free fall: \( h = \frac{1}{2} g t^{2} \).

Step 2: Solve for Time (a)

In the equation \( h = \frac{1}{2} g t^{2} \), set \( h = 3.00 \, \text{m} \) and \( g = 9.81 \, \text{m/s}^2 \). Solve for \( t \):\[ 3.00 = \frac{1}{2} \times 9.81 \times t^{2} \]\[ t^{2} = \frac{3.00 \times 2}{9.81} \]\[ t = \sqrt{\frac{6}{9.81}} \approx 0.782 \, \text{s} \]

Step 3: Determine Speed Just Before Impact (b)

Find the speed of the bolt just before it hits the elevator floor using \( v = gt \). Since the initial velocity is 0 m/s (relative to elevator), \( g = 9.81 \, \text{m/s}^2 \), and \( t \approx 0.782 \, \text{s} \):\[ v = 9.81 \times 0.782 \approx 7.67 \, \text{m/s} \]

Step 4: Account for Speed Relative to Elevator (b)

The observer in the elevator sees the bolt falling due to gravity without considering the elevator's own motion, so the speed remains \( 7.67 \, \text{m/s} \).

Step 5: Analyze from Building Observer's Perspective (c)

The observer in the building sees the elevator moving up at \( 2.50 \, \text{m/s} \). When falling from the ceiling of a moving elevator, the initial velocity of the bolt relative to the ground is \( -2.50 \, \text{m/s} \). Use \( v = v_0 + gt \) where \( v_0 = -2.50 \, \text{m/s}, g = 9.81 \, \text{m/s}^2, \) and \( t \approx 0.782 \, \text{s} \):\[ v = -2.50 + 9.81 \times 0.782 \approx 5.17 \, \text{m/s} \]

Step 6: Calculate Total Distance Travelled (d)

The observer in the building calculates the distance not by the height (3.00 m) but the path traveled. Since the bolt started with a downward initial velocity relative to the ground, use \( s = v_0 t + \frac{1}{2} g t^2 \):\[ s = -2.50 \times 0.782 + \frac{1}{2} \times 9.81 \times (0.782)^2 \approx 1.96 \, \text{m} \]

Key Concepts

Free Fall

Understanding free fall is crucial when discussing projectile motion and objects accelerating under gravity. In this context, free fall refers to the motion of an object subject only to the force of gravity. It is a key element in solving the problem with the falling bolt. Here, the bolt in the elevator falls due to gravity alone, after it becomes loose from the ceiling.
The problem assumes the absence of air resistance and the action of gravity at a standard value of 9.81 m/s². It’s important to remember this concept is the same whether the observer is inside the elevator or outside on a building landing. Here’s a quick summary to remember about free fall:

• Objects in free fall experience uniform acceleration due to gravity.
• Initial velocity in free fall can differ based on the reference frame (as seen in this exercise).
• Equations like \( h = \frac{1}{2} g t^2\) help determine time or distance during a fall.

These principles will guide further calculations of velocity at impact and how gravity acts equally in all scenarios.

Reference Frames

Reference frames are an essential concept in understanding how observed motion can differ. In the context of our exercise, two major reference frames are of interest: the moving elevator and the stationary building observer. Reference frames are crucial because they influence our perception of the object's velocity and motion. Let's break this concept down:

• In physics, a reference frame is a perspective from which observation and measurements are made.
• Each observer (inside and outside the elevator) perceives the bolt’s motion differently due to differing velocities of their frames.
• In the elevator’s frame, the ceiling and floor move together, so only gravity affects the bolt.

For students, it’s vital to grasp that resolving motions in different frames requires adjusting the initial conditions and understanding relativity in velocity results.

Kinematics

Kinematics deals with the motion of objects without considering the forces that cause such motion. It gives us the tools to calculate positions, velocities, and accelerations. In this problem, kinematics helps solve how long the bolt takes to hit the floor and its speed upon impact in various frames. Vivid understanding of kinematics involves:

• Recognizing that constant acceleration (like gravity in free fall) leads to specific kinematic equations.
• Using equations: \( v = gt \) and \( h = \frac{1}{2} g t^2 \) to relate time, velocity, and distance.
• Understanding initial conditions for velocity and position, especially address changes as seen from different reference frames.

Kinematic principles ensure clarity when transitioning between scenarios such as those the two observers experience.

Relative Velocity

Relative velocity is the velocity of one object as observed from another moving object. In this exercise, calculating relative velocity helps observers determine the bolt's speed from different viewpoints. Here's why understanding relative velocity matters:

• It's about how fast one object appears to move from another object's viewpoint.
• The elevator observer sees the bolt solely in free fall, so the relative velocity doesn’t include upward motion.
• The building observer, however, sees the bolt with an initial relative velocity opposite to the elevator's upward movement (i.e., -2.50 m/s).

It's crucial to calculate velocities correctly and understand the interplay between motion of different frames. These analyses give varying velocities as experienced by different observers in the scenario, providing comprehensive insight into movement dynamics.