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title: "Three objects move along a straight line in the positive x-direction. Each object starts from rest at the origin at time \\(t = 0\\) and reaches a speed \\(v_0\\) at time \\(t = T\\). During the time interval \\(0 \\le t \\le T\\), the velocities of the three objects as functions of time are given by:  Object 1: \\(v_1(t) = v_0 \\left(\\dfrac{t}{T}\\right)\\)  Object 2: \\(v_2(t) = v_0 \\left(\\dfrac{t}{T}\\right)^2\\)  Object 3: \\(v_3(t) = v_0 \\left(\\dfrac{t}{T}\\right)^{1/2}\\)  Which of the following correctly ranks the displacements \\(\\Delta x_1\\), \\(\\Delta x_2\\), and \\(\\Delta x_3\\) of the objects from \\(t = 0\\) to \\(t = T\\)?"
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url: "https://nerd-notes.com/ubq/123956/"
date_modified: "2026-09-28T13:27:36+00:00"
---

# Three objects move along a straight line in the positive x-direction. Each object starts from rest at the origin at time \(t = 0\) and reaches a speed \(v_0\) at time \(t = T\). During the time interval \(0 \le t \le T\), the velocities of the three objects as functions of time are given by:

Object 1: \(v_1(t) = v_0 \left(\dfrac{t}{T}\right)\)

Object 2: \(v_2(t) = v_0 \left(\dfrac{t}{T}\right)^2\)

Object 3: \(v_3(t) = v_0 \left(\dfrac{t}{T}\right)^{1/2}\)

Which of the following correctly ranks the displacements \(\Delta x_1\), \(\Delta x_2\), and \(\Delta x_3\) of the objects from \(t = 0\) to \(t = T\)?

Three objects move along a straight line in the positive x-direction. Each object starts from rest at the origin at time \(t = 0\) and reaches a speed \(v_0\) at time \(t = T\). During the time interval \(0 \le t \le T\), the velocities of the three objects as functions of time are given by:

Object 1: \(v_1(t) = v_0 \left(\dfrac{t}{T}\right)\)

Object 2: \(v_2(t) = v_0 \left(\dfrac{t}{T}\right)^2\)

Object 3: \(v_3(t) = v_0 \left(\dfrac{t}{T}\right)^{1/2}\)

Which of the following correctly ranks the displacements \(\Delta x_1\), \(\Delta x_2\), and \(\Delta x_3\) of the objects from \(t = 0\) to \(t = T\)?

- **A.** \(\Delta x_3 > \Delta x_1 > \Delta x_2\)
- **B.** \(\Delta x_2 > \Delta x_1 > \Delta x_3\)
- **C.** \(\Delta x_1 > \Delta x_3 > \Delta x_2\)
- **D.** \(\Delta x_1 = \Delta x_2 = \Delta x_3\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/123956/*