Question

In a fancy hotel, the back of the elevator is made of glass so that you can enjoy a lovely view on your ride. The elevator travels at an average speed of \(1.75 \mathrm{~m} / \mathrm{s}\). A boy on the 15th floor, \(80.0 \mathrm{~m}\) above the ground level, drops a rock at the same instant the elevator starts its ascent from the 1st to the 5th floor. Assume the elevator travels at its average speed for the entire trip and neglect the dimensions of the elevator. a) How long after it was dropped do you see the rock? b) How long does it take for the rock to reach ground level?

Short answer

b) What is the time it takes for the rock to reach the ground level?

Step-by-step solution

Calculate initial height difference

We are given that the boy is on the 15th floor, which is \(80.0 \mathrm{~m}\). The elevator is at the ground level, so the initial height difference is: \(h_{initial} = 80.0 \mathrm{~m}\) **Step 2: Calculate the moment the rock becomes visible to the elevator**

Determine when the rock is visible

For the rock to become visible to the elevator, we have to equate their heights: \(h_{elevator} = h_{rock}\) \(v_{elevator} \cdot t_{visible} = 80.0 \mathrm{~m} - \frac{1}{2}g \cdot t_{visible}^2\) Plugging the given value of \(v_{elevator} = 1.75 \mathrm{~m} / \mathrm{s}\), and \(g = 9.8 \mathrm{~m} / \mathrm{s}^2\), we get: \(1.75 \mathrm{~m} / \mathrm{s} \cdot t_{visible} = 80 \mathrm{~m} - \frac{1}{2}(9.8 \mathrm{~m} / \mathrm{s}^2) \cdot t_{visible}^2\) Carefully solve this equation for \(t_{visible}\), which will give you the time after which the rock becomes visible to the elevator. **Step 3: Calculate the time it takes for the rock to reach ground level**

Determine the time for the rock to reach ground level

To find the time it takes for the rock to reach the ground level, we use the equation of motion for the rock: \(d_{rock} = 80.0\mathrm{~m} - \frac{1}{2}g \cdot t_{ground}^2\) Plugging the values, we get: \(0 = 80\mathrm{~m} - \frac{1}{2}(9.8\mathrm{~m} / \mathrm{s}^2) \cdot t_{ground}^2\) Carefully solve this equation for \(t_{ground}\), which will give you the time it takes for the rock to reach ground level.

Key concepts

Kinematics

Kinematics is a fundamental concept in physics that deals with the motion of objects without considering the forces causing the motion. When analyzing problems like the one with the rock and elevator, it's all about understanding the relationship between distance, speed, and time.

In this problem, kinematics helps us to figure out how long it takes for the rock to fall to ground level and how long it takes until it is visible by the elevator. We break down the movement into segments and use kinematic equations to solve for time.

• For the elevator, constant speed means it travels a specific distance at a steady pace.
• For the rock, we consider its journey from being stationary to accelerating under gravity's influence.

By understanding these principles, you can dissect motion into manageable parts and predict the behavior of moving objects.

Equations of Motion

Equations of motion are essential tools in kinematics. They allow us to calculate various parameters of a moving object, such as displacement, velocity, and time. Let's take a deeper look at how they are applied in scenarios like the rock drop from the 15th floor.

In this exercise, the second equation of motion is crucial:\[ d = vt + \frac{1}{2}at^2 \]This equation helps us establish the position of the rock as a function of time and solve for when it reaches the same height as the elevator or when it reaches the ground. Consider each part:

• The term \(vt\) accounts for constant velocity, which in this case, corresponds to any initial velocity.
• The term \(\frac{1}{2}at^2\) deals with acceleration acting on the moving object, notably the gravitational pull on the rock (i.e., \(g = 9.8 \mathrm{~m/s^2}\)).

Solver techniques such as substitution are then used to determine the interaction point or when one movement overtakes another.

Free Fall

Free fall describes the motion of objects falling solely under the influence of gravity, with air resistance deemed negligible. In the scenario described, the rock dropped from the 15th floor is in free fall.

One of the key features of free fall is that the only force acting on the object is gravity, which simplifies analysis through precise equations. The rock experiences continuous acceleration downward due to gravity at \(9.8 \mathrm{~m/s^2}\). For such a case:

• Initial velocity is zero when the object starts from rest.
• Acceleration is constant and equivalent to the gravitational acceleration \(g\).

These principles allow us to use the motion equation to determine how long it takes the rock to hit the ground. Knowing the initial height allows us to connect time and distance using the second equation of motion, isolating time as the subject, ultimately calculating how short or long it will take for the rock's brisk descent to complete.