Problem 11
Question
A certain escalator travels at a rate of \(10.6 \mathrm{m} / \mathrm{min}\), and its angle of inclination is \(32.5^{\circ} .\) What is the vertical component of the velocity? How long will it take a passenger to travel 10.0 m vertically?
Step-by-Step Solution
Verified Answer
The vertical component of the velocity is approximately \(5.59\,\mathrm{m/min}\). To travel 10.0 m vertically, it takes approximately 1.79 minutes.
1Step 1: Calculating the Vertical Component of Velocity
To find the vertical component of the velocity, use the formula for the vertical component which is the product of the full velocity and the sine of the angle of inclination: \(v_{vertical} = v_{total} \cdot \sin(\theta)\). Here, \(v_{total} = 10.6 \mathrm{m} / \mathrm{min}\) and \(\theta = 32.5^\circ\).
2Step 2: Determine Time to Travel a Vertical Distance
The time taken to travel a certain vertical distance can be calculated using the formula: \(t = \frac{d_{vertical}}{v_{vertical}}\), where \(d_{vertical} = 10.0\,m\) is the vertical distance, and \(v_{vertical}\) is the vertical component of velocity found in Step 1.
Key Concepts
Trigonometric Functions in VelocityCalculating Velocity ComponentsTime and Distance Calculation
Trigonometric Functions in Velocity
When analyzing motion, it's crucial to understand how to decompose velocity into its horizontal and vertical components. This is especially important when dealing with motion on an incline, such as an escalator or a projectile's path. Trigonometric functions allow us to calculate these components efficiently.
For a given velocity vector, the usage of sine and cosine from trigonometry becomes a practical tool. In the context of an escalator, the angle of inclination gives a clear relationship between the total velocity and its vertical and horizontal components. The sine function, in particular, relates the angle of inclination to the vertical component of velocity.
To visualize this, imagine a right triangle where the hypotenuse represents the total velocity of the escalator, and the angle between the hypotenuse and the base is the angle of inclination. The vertical side of the triangle represents the vertical component of velocity. Through trigonometric functions, we can deduce that the length of this vertical side, which is what we're interested in, is the product of the hypotenuse (total velocity) and the sine of the angle of inclination.
For a given velocity vector, the usage of sine and cosine from trigonometry becomes a practical tool. In the context of an escalator, the angle of inclination gives a clear relationship between the total velocity and its vertical and horizontal components. The sine function, in particular, relates the angle of inclination to the vertical component of velocity.
To visualize this, imagine a right triangle where the hypotenuse represents the total velocity of the escalator, and the angle between the hypotenuse and the base is the angle of inclination. The vertical side of the triangle represents the vertical component of velocity. Through trigonometric functions, we can deduce that the length of this vertical side, which is what we're interested in, is the product of the hypotenuse (total velocity) and the sine of the angle of inclination.
Calculating Velocity Components
Calculating the vertical and horizontal components of velocity is foundational in understanding motion in two dimensions. In our case, with the escalator moving at an angle to the vertical, we're particularly concerned with how quickly it moves up, which is its vertical component of velocity.
The formula to obtain this vertical velocity component is straightforward: \(v_{vertical} = v_{total} \cdot \sin(\theta)\), where \(v_{total}\) is the total velocity of the escalator, and \(\theta\) is the angle of inclination. By plugging in the values from the problem—\(v_{total} = 10.6\,\text{m/min}\) and \(\theta = 32.5^\circ\)—we apply this formula to obtain the vertical component.
This step is crucial, as it separates the effect of velocity along the direction we're interested in from the total motion. It simplifies further calculations regarding time and distance traveled, ensuring accurate results when solving real-world problems.
The formula to obtain this vertical velocity component is straightforward: \(v_{vertical} = v_{total} \cdot \sin(\theta)\), where \(v_{total}\) is the total velocity of the escalator, and \(\theta\) is the angle of inclination. By plugging in the values from the problem—\(v_{total} = 10.6\,\text{m/min}\) and \(\theta = 32.5^\circ\)—we apply this formula to obtain the vertical component.
This step is crucial, as it separates the effect of velocity along the direction we're interested in from the total motion. It simplifies further calculations regarding time and distance traveled, ensuring accurate results when solving real-world problems.
Time and Distance Calculation
Once the vertical component of velocity is determined, calculating the time it takes to travel a certain vertical distance is the next logical step. This involves a simple formula that relates distance, velocity, and time: \(t = \frac{d_{vertical}}{v_{vertical}}\), where \(t\) is the time, \(d_{vertical}\) is the vertical distance to be traveled, and \(v_{vertical}\) is the vertical component of velocity.
By substituting the obtained vertical velocity into this formula, we can find the time it would take a passenger to move up a specific distance vertically. For example, if a passenger was to travel 10.0 meters up the escalator, and we've already determined the escalator's vertical velocity component, we can quickly find how long the journey will take. This is crucial for planning and can have applications ranging from the design of transportation systems to predicting the time of arrival in different scenarios.
Understanding how to manipulate these formulas and concepts is not just about solving an exercise; it helps students appreciate the underlying mechanics of motion in real-world contexts, reinforcing their overall comprehension of physics and mathematics.
By substituting the obtained vertical velocity into this formula, we can find the time it would take a passenger to move up a specific distance vertically. For example, if a passenger was to travel 10.0 meters up the escalator, and we've already determined the escalator's vertical velocity component, we can quickly find how long the journey will take. This is crucial for planning and can have applications ranging from the design of transportation systems to predicting the time of arrival in different scenarios.
Understanding how to manipulate these formulas and concepts is not just about solving an exercise; it helps students appreciate the underlying mechanics of motion in real-world contexts, reinforcing their overall comprehension of physics and mathematics.
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