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Chapter 2: Problem 75

A speed ramp at an airport is basically a large conveyor belt on which you can stand and be moved along. The belt of one ramp moves at a constant speed such that a person who stands still on it leaves the ramp \(64 \mathrm{s}\) after getting on. Clifford is in a real hurry, however, and skips the speed ramp. Starting from rest with an acceleration of \(0.37 \mathrm{m} / \mathrm{s}^{2}\), he covers the same distance as the ramp does, but in one-fourth the time. What is the speed at which the belt of the ramp is moving?

Short Answer

Expert verified
The speed of the ramp is 0.74 m/s.

Step by step solution

01

Determine the Distance Covered by the Ramp

First, find the distance the ramp covers. We know it takes 64 s to travel this distance when standing.The formula for distance when speed is constant is:\[ d = v \cdot t \]Let's assume the speed of the ramp is \( v_r \). So, the distance \( d \) covered by the ramp is:\[ d = v_r \times 64 \]
02

Determine Time Taken by Clifford

Clifford covers the same distance in one-fourth the time taken by the belt. Since the ramp takes 64 s, Clifford's time is:\[ t_c = \frac{64}{4} = 16 \text{ s} \]
03

Use the Kinematic Equation for Clifford

Clifford starts from rest and accelerates at \(0.37 \mathrm{m/s^2}\) over 16 seconds. We can use the kinematic equation to find the distance he covers:\[ d = ut + \frac{1}{2} a t^2 \]where \( u = 0 \) (starting from rest), \( a = 0.37 \mathrm{m/s^2} \), and \( t = 16 \text{ s}.\)Plug in the values:\[ d = 0 + \frac{1}{2} \cdot 0.37 \cdot (16)^2 \]\[ d = 0.185 \cdot 256 \]\[ d = 47.36 \text{ m} \]
04

Determine Ramp Speed

Now that we have the same distance \( d = 47.36 \text{ m} \) for the ramp, use the equation:\[ d = v_r \cdot 64 \]Substitute \( d = 47.36 \) m to solve for \( v_r \):\[ 47.36 = v_r \cdot 64 \]\[ v_r = \frac{47.36}{64} \]\[ v_r = 0.74 \text{ m/s} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Acceleration
Acceleration is a key concept in kinematics. It describes how quickly an object changes its speed over time. In physics, acceleration is often measured in meters per second squared (m/s虏). A positive acceleration means an increase in speed, while a negative acceleration (often called deceleration) indicates a reduction in speed.

For example, in the exercise, Clifford accelerates from rest with an acceleration of 0.37 m/s虏. This means that each second, his speed increases by 0.37 meters per second. Acceleration is crucial to understanding how quickly objects can reach a specific speed and travel a certain distance.
Distance
Distance is a measure of how far an object travels over a period of time. In kinematic problems, it is often crucial to determine this distance to understand motion better. Distance is a scalar quantity, meaning it only has magnitude and no direction. It is typically measured in meters (m).

In the exercise, both Clifford and the speed ramp cover a distance, which is calculated using different methods. For Clifford, who accelerates, the distance is calculated using the kinematic equation for uniformly accelerated motion. The ramp's distance is calculated based on constant speed over time. Understanding how to calculate distance in different scenarios is key to mastering kinematic problems.
Speed
Speed refers to how fast an object is moving. It is a scalar quantity, which means it has magnitude but no direction. Speed measures how much distance an object covers within a specific period, typically expressed in meters per second (m/s).
  • Constant speed: When an object travels the same distance every second.
  • Varied speed: When the speed changes over time, we take the average speed over the period.
In our example, the speed ramp moves at a constant speed. By calculating the speed, we found that the ramp's speed is 0.74 m/s. Understanding speed is critical for predicting how long it will take an object to travel a certain distance or for determining which factors will affect travel time.
Kinematic Equations
Kinematic equations are essential in solving problems related to motion. They provide the mathematical relationships between the main variables in motion: acceleration, initial speed, final speed, distance, and time. Each equation applies to uniformly accelerated motion, where the acceleration remains constant.In the exercise problem, one of the kinematic equations \[ d = ut + \frac{1}{2} a t^2 \]is used to calculate the distance Clifford covers because he starts at rest and accelerates. Here:
  • \( u \) is the initial velocity (0 m/s in this case).
  • \( a \) is the acceleration (0.37 m/s虏).
  • \( t \) is the time (16 s).
By using these equations, we can solve for various factors in motion problems, making them a powerful tool for analyzing kinematics.

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Most popular questions from this chapter

While standing on a bridge \(15.0 \mathrm{m}\) above the ground, you drop a stone from rest. When the stone has fallen \(3.20 \mathrm{m},\) you throw a second stone straight down. What initial velocity must you give the second stone if they are both to reach the ground at the same instant? Take the downward direction to be the negative direction.

You step onto a hot beach with your bare feet. A nerve impulse, generated in your foot, travels through your nervous system at an average speed of \(110 \mathrm{m} / \mathrm{s}\). How much time does it take for the impulse, which travels a distance of \(1.8 \mathrm{m},\) to reach your brain?

A train has a length of 92 m and starts from rest with a constant acceleration at time \(t=0\) s. At this instant, a car just reaches the end of the train. The car is moving with a constant velocity. At a time \(t=14 \mathrm{s}\), the car just reaches the front of the train. Ultimately, however, the train pulls ahead of the car, and at time \(t=28 \mathrm{s},\) the car is again at the rear of the train. Find the magnitudes of (a) the car's velocity and (b) the train's acceleration.

A diver springs upward with an initial speed of \(1.8 \mathrm{m} / \mathrm{s}\) from a \(3.0-\mathrm{m}\) board. (a) Find the velocity with which he strikes the water. / Hint: When the diver reaches the water, his displacement is \(y=-3.0 \mathrm{m}\) (measured from the board), assuming that the dowmward direction is chosen as the negative direction./ (b) What is the highest point he reaches above the water?

A hot-air balloon is rising upward with a constant speed of \(2.50 \mathrm{m} / \mathrm{s}\). When the balloon is \(3.00 \mathrm{m}\) above the ground, the balloonist accidentally drops a compass over the side of the balloon. How much time elapses before the compass hits the ground?
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