---
title: "Two small spheres, A and B, are launched horizontally from the edges of two different tables. The launch conditions for each sphere are provided in the table below.  | Sphere | Launch Speed | Launch Height | | :— | :— | :— | | A | \\(v_0\\) | \\(H\\) | | B | \\(2v_0\\) | \\(H/2\\) |  Let \\(\\Delta t_A\\) and \\(R_A\\) be the time of flight and the horizontal range of sphere A, respectively. Which of the following correctly relates the time of flight \\(\\Delta t_B\\) and the horizontal range \\(R_B\\) of sphere B to those of sphere A?"
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url: "https://nerd-notes.com/ubq/113920/"
date_modified: "2026-06-22T07:38:24+00:00"
---

# Two small spheres, A and B, are launched horizontally from the edges of two different tables. The launch conditions for each sphere are provided in the table below.

| Sphere | Launch Speed | Launch Height |
| :— | :— | :— |
| A | \(v_0\) | \(H\) |
| B | \(2v_0\) | \(H/2\) |

Let \(\Delta t_A\) and \(R_A\) be the time of flight and the horizontal range of sphere A, respectively. Which of the following correctly relates the time of flight \(\Delta t_B\) and the horizontal range \(R_B\) of sphere B to those of sphere A?

Two small spheres, A and B, are launched horizontally from the edges of two different tables. The launch conditions for each sphere are provided in the table below.

| Sphere | Launch Speed | Launch Height |
| :--- | :--- | :--- |
| A | \(v_0\) | \(H\) |
| B | \(2v_0\) | \(H/2\) |

Let \(\Delta t_A\) and \(R_A\) be the time of flight and the horizontal range of sphere A, respectively. Which of the following correctly relates the time of flight \(\Delta t_B\) and the horizontal range \(R_B\) of sphere B to those of sphere A?

![Two diagrams side-by-side. On the left, sphere A is launched from a table of height H with a horizontal velocity vector labeled v_0. On the right, sphere B is launched from a table of height H/2 (half as tall) with a horizontal velocity vector labeled 2v_0. Both spheres have dashed parabolic paths leading to a horizontal floor.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1782113904-duh7kw.jpg)

- **A.** | Time of Flight | Horizontal Range | | :--- | :--- | | \(\Delta t_B = \dfrac{\Delta t_A}{\sqrt{2}}\) | \(R_B = \sqrt{2} R_A\) |
- **B.** | Time of Flight | Horizontal Range | | :--- | :--- | | \(\Delta t_B = \dfrac{\Delta t_A}{2}\) | \(R_B = R_A\) |
- **C.** | Time of Flight | Horizontal Range | | :--- | :--- | | \(\Delta t_B = \dfrac{\Delta t_A}{\sqrt{2}}\) | \(R_B = 2 R_A\) |
- **D.** | Time of Flight | Horizontal Range | | :--- | :--- | | \(\Delta t_B = \Delta t_A\) | \(R_B = 2 R_A\) |

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/113920/*