Problem 97 - Precalculus

Question

Find the linear speed of a point that moves with constant speed in a circular motion if the point travels along the circle of are length \(s\) in time \(t\). $$s=68,000 \mathrm{km}, t=250 \mathrm{hr}$$

Step-by-Step Solution

Answer

The linear speed is 272 km/hr.

Step 1: Understanding the Problem

We need to find the linear speed of a point moving along a circle. The problem provides the arc length \( s = 68,000 \, \text{km} \) that the point travels, and the time \( t = 250 \, \text{hr} \) it takes to travel that distance.

Step 2: Formula for Linear Speed

The linear speed \( v \) of a point moving along a path is given by the formula \( v = \frac{s}{t} \), where \( s \) is the arc length traversed and \( t \) is the time taken.

Step 3: Substitute Known Values

Substitute \( s = 68,000 \, \text{km} \) and \( t = 250 \, \text{hr} \) into the formula: \[ v = \frac{68,000}{250} \]

Step 4: Calculate the Linear Speed

Perform the division: \[ v = \frac{68,000}{250} = 272 \, \text{km/hr} \]

Step 5: Interpretation of Result

The linear speed of the point moving along the circle is \( 272 \, \text{km/hr} \), meaning that the point covers 272 kilometers every hour that it is in motion.

Key Concepts

Circular Motion

Circular motion describes the movement of a point or object that travels along a circular path. This type of motion is common in daily life, such as the movement of the hands of a clock or a car steering around a curve. A characteristic feature of circular motion is the continual change in direction, even if the speed remains constant. This is because the object is moving in a curve, constantly changing its orientation relative to a center point.

In circular motion, it's important to distinguish between linear speed and angular speed. Linear speed relates to how fast the object is moving along the circumference of the circle, whereas angular speed relates to how quickly the object sweeps through an angle as it moves along that path. These two concepts, though different, are interrelated through the geometry of the circle.

Arc Length

The arc length is a measure of the distance along the curved path of a circle between two points. It is basically what you would measure if you were to "unwrap" the circle into a straight path. Arc length is often denoted by the symbol \( s \) and can be calculated if you know the radius and angle in radians, although in some cases, like this exercise, it is provided directly.

The formula to calculate arc length \( s \) when you know the radius \( r \) and the angle \( \theta \) in radians is:

• \( s = r \times \theta \)

However, in this problem, you already have an arc length, which makes it simpler to focus directly on finding the linear speed using this measure. Understanding arc length helps in identifying how much of the circle's circumference is being traversed by the point.

Time Calculation

Time calculation in the context of motion along a circle refers to how long it takes for a point or object to complete a certain distance along the circular path. It is a crucial variable because, together with arc length, it allows us to find the linear speed of the moving point.

In our exercise, the total time \( t \) given is 250 hours. This is the duration over which the point traverses the arc length of 68,000 km. Having accurate time measurements is critical because they directly affect how we assess speed, providing a frame to measure how fast or slow the point is moving.

Knowing the time helps to determine whether the movement is likely to be sustained at a constant speed or if there might be accelerations or decelerations, though the latter are not considered in this particular problem.

Motion along a Circle

Motion along a circle involves understanding the dynamics of movement in a circular pattern. When we deal with motion along a circle, we are interested in how the point or object moves around the perimeter of the circle, which remains equidistant from the center.

This type of motion can be influenced by various forces, such as gravitational or centripetal force, which keep the object in its path. However, for the sake of our simple calculation, the focus is on one-dimensional motion around the circle - essentially tracking a single point's journey along the perimeter.

To determine linear speed while considering motion along a circle, you calculate how much distance is covered (arc length) over a given period (time). This is expressed in the formula \( v = \frac{s}{t} \), where \( v \) is the linear speed, \( s \) is arc length, and \( t \) is time. Applying this to the exercise, we found the speed as 272 km/hr. This underscores how motion along a circle is quantitatively described using linear speed ensuring both concepts are well aligned.