Question

Think \& Calculate A child rides a pony on a circular track whose radius is \(4.5 \mathrm{~m}\). (a) Find the distance traveled and the displacement after the child has gone halfway around the track. (b) Has the distance traveled increased, decreased, or stayed the same when the child has completed one circuit of the track? Explain. (c) Has the displacement increased, decreased, or stayed the same when the child has completed one circuit of the track? Explain. (d) Find the distance and displacement after a complete circuit of the track.

Short answer

(a) Distance: \(4.5\pi\) m; Displacement: 9 m. (b) Increased. (c) Decreased to zero. (d) Distance: \(9\pi\) m; Displacement: 0 m.

Step-by-step solution

Understand the Problem

We have a circular track with radius \( r = 4.5 \, \text{m} \). The child rides halfway and then completes a full circle. We need to calculate the distance traveled and the displacement for both scenarios.

Find Distance for Halfway Around Track

The circumference of the circle is given by \( C = 2\pi r \). Calculate the distance for halfway around using the formula: \[ \text{Distance for half circle} = \frac{1}{2} C = \frac{1}{2} (2\pi \cdot 4.5) = 4.5\pi \].

Calculate Displacement for Halfway

Displacement is the straight-line distance from the start point to end point. For halfway, the displacement is the diameter of the circle. \( \text{Diameter} = 2r = 2 \times 4.5 = 9 \text{ m} \).

Discuss Distance Change After Full Circle

After completing a full circle, the distance traveled is equal to the circumference, which is \[ 2\pi r = 9\pi \]. The distance has doubled compared to halfway, therefore the distance traveled has increased.

Discuss Displacement Change After Full Circle

After completing a full circle, the child ends up at the starting point. Therefore, the displacement is \( 0 \text{ m} \). Compared to the halfway scenario where displacement is \( 9 \text{ m} \), the displacement has decreased to zero.

Find Distance and Displacement for Full Circuit

The complete distance for a full circuit after traveling around the track is \( 9\pi \) meters. The displacement after a full circuit is \( 0 \text{ m} \). Therefore, the distance traveled is \( 9\pi \) meters and the displacement is \( 0 \text{ m} \).

Key concepts

Understanding Distance in Circular Motion

Distance is a measure of the total path traveled by an object, regardless of the direction. It is a scalar quantity, meaning it only has magnitude and no direction. In the case of a circular track, the distance traveled is based on the circumference of the circle.

• The formula for the circumference of a circle is given by: \( C = 2\pi r \).
• For the scenario where the child has gone halfway around the track, the distance traveled is half the circumference: \( \frac{1}{2} C = \frac{1}{2} \times 2\pi \times 4.5 = 4.5\pi \text{ meters} \).
• When the child completes one full circuit of the track, the distance traveled is the complete circumference, which is \( 9\pi \text{ meters} \).

It's important to notice that the distance increases as the path traveled increases. Thus, after a full circle, even if the displacement becomes zero (as we'll discuss later), the distance definitely increases.

Exploring Displacement in Circular Motion

Displacement is a vector quantity, which measures the change in position of an object. Unlike distance, displacement has both magnitude and direction. In circular motion, displacement reflects the straight-line distance from the starting point to the endpoint.

• For halfway around the circle, the child's displacement equals the diameter of the circle. This is because the start point and the endpoint form a straight line through the circle's center: \( \text{Displacement} = 2r = 9 \text{ meters} \).
• If the child completes a full circuit, they return to the starting point. As a result, the displacement becomes \( 0 \text{ meters} \) since the overall change in position is zero.

From halfway to a full cycle, while the distance increases to encompass the full circumference, the displacement decreases to zero, showing the distinct nature of these two concepts.

Role of Radius in a Circular Path

The radius of a circular track is crucial in determining both the distance and displacement. The radius, denoted as \( r \), is the distance from the center of the circle to any point on its perimeter. It is the backbone for calculating important circular motion parameters.

• The radius directly feeds into calculating the circumference (hence the distance), utilizing the formula \( C = 2\pi r \).
• The radius also influences the displacement when considering halfway around the circle. Here, the displacement is double the length of the radius (the diameter), which gives \( 2r \).

In this exercise, with a radius of \( 4.5 \text{ meters} \), these measurements reveal both distance and displacement calculations. The radius fundamentally shapes the path and the dynamics of the motion along the circular track.