---
title: "A cart moves along a level, horizontal track with a constant acceleration \\(a\\) directed in the \\(+x\\)-direction. At time \\(t = 0\\), the cart passes the origin \\(x = 0\\) with velocity \\(v_0 \\hat{i}\\), and simultaneously launches a small ball vertically upward with speed \\(u_0\\) relative to the cart. Air resistance is negligible, and the downward acceleration due to gravity is \\(g\\). Which of the following correctly gives the trajectory equation \\(y(x)\\) of the ball in the ground reference frame (for \\(y \\ge 0\\)) and the position of the ball relative to the launcher, \\(\\Delta x = x_{\\text{ball}} – x_{\\text{cart}}\\), at the instant the ball returns to its launch height?"
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url: "https://nerd-notes.com/ubq/124129/"
date_modified: "2026-09-28T13:30:36+00:00"
---

# A cart moves along a level, horizontal track with a constant acceleration \(a\) directed in the \(+x\)-direction. At time \(t = 0\), the cart passes the origin \(x = 0\) with velocity \(v_0 \hat{i}\), and simultaneously launches a small ball vertically upward with speed \(u_0\) relative to the cart. Air resistance is negligible, and the downward acceleration due to gravity is \(g\). Which of the following correctly gives the trajectory equation \(y(x)\) of the ball in the ground reference frame (for \(y \ge 0\)) and the position of the ball relative to the launcher, \(\Delta x = x_{\text{ball}} – x_{\text{cart}}\), at the instant the ball returns to its launch height?

A cart moves along a level, horizontal track with a constant acceleration \(a\) directed in the \(+x\)-direction. At time \(t = 0\), the cart passes the origin \(x = 0\) with velocity \(v_0 \hat{i}\), and simultaneously launches a small ball vertically upward with speed \(u_0\) relative to the cart. Air resistance is negligible, and the downward acceleration due to gravity is \(g\). Which of the following correctly gives the trajectory equation \(y(x)\) of the ball in the ground reference frame (for \(y \ge 0\)) and the position of the ball relative to the launcher, \(\Delta x = x_{\text{ball}} - x_{\text{cart}}\), at the instant the ball returns to its launch height?

![A schematic diagram showing a wheeled cart on a flat, horizontal surface aligned with a horizontal axis labeled x. The cart is rendered as a rectangle resting on two small circular wheels. An arrow pointing horizontally to the right above the cart is labeled v_0, and another horizontal arrow pointing to the right is labeled a. Mounted at the center of the cart's upper deck is a vertical launching tube, from which a small circular ball emerges. A vertical dashed arrow pointing directly upward from the ball is labeled u_0. To the side of the diagram, a downward vertical arrow is labeled g to indicate the direction of free fall. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790602236-GcIH1Q.jpg)

- **A.** \(y(x) = \dfrac{u_0}{v_0}x - \dfrac{\sqrt{g^2 + a^2}}{2v_0^2}x^2\quad\text{and}\quad\Delta x = -\dfrac{a u_0^2}{2g^2}\)
- **B.** \(y(x) = \dfrac{u_0}{v_0}x - \dfrac{g}{2v_0^2}x^2\quad\text{and}\quad\Delta x = -\dfrac{a u_0^2}{2g^2}\)
- **C.** \(y(x) = \dfrac{u_0}{v_0}x - \dfrac{g}{2v_0^2}x^2\quad\text{and}\quad\Delta x = -\dfrac{2 a u_0^2}{g^2}\)
- **D.** \(y(x) = \dfrac{u_0}{v_0}x - \dfrac{\sqrt{g^2 + a^2}}{2v_0^2}x^2\quad\text{and}\quad\Delta x = -\dfrac{2 a u_0^2}{g^2}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124129/*