**Modeling 1-D Straight Line Motion - Educational Example**A basketball is dropped from a walkway above Bartow Arena. It falls from rest with a constant acceleration of 9.8 m/s² until it reaches the floor 2.5 seconds later. A physics class is modeling this 1-D straight line motion as part of a class project. They want to construct a simple, but accurate, data table using the following equations:\[ v_{i+1} = v_i + a \cdot t \]\[ x_{i+1} = x_i + v_i \cdot t \]Their instructor requires them to make the constant time interval \(\Delta t\) between rows in their table sufficiently small so that they have only a 4 percent error in the total distance traveled by the ball during the 2.5 seconds. **Question:** What should be the value of \(\Delta t\) in their data table to cover the entire time interval and achieve the desired error?### Explanation#### Equations of Motion:1. **Velocity Update:** \[ v_{i+1} = v_i + a \cdot t \] Where: - \( v_{i+1} \) is the velocity at the next time step. - \( v_i \) is the current velocity. - \( a \) is the constant acceleration (9.8 m/s²). - \( t \) is the time interval.2. **Position Update:** \[ x_{i+1} = x_i + v_i \cdot t \] Where: - \( x_{i+1} \) is the position at the next time step. - \( x_i \) is the current position. - \( v_i \) is the current velocity. - \( t \) is the time interval.#### Task:To determine the appropriate time interval \(\Delta t\) so that the error in the total distance traveled by the basketball is within 4 percent over the 2.5-second fall.By calculating and adjusting \(\Delta t\), we ensure the data table provides a precise representation of the basketball’s fall, crucial for the class project’s accuracy. This example showcases fundamental physics concepts like constant acceleration, kinematic equations, and the importance of precision in experimental physics.

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A basketball is dropped from a walkway above Bartow Arena. It falls from rest with a constant acceleration of 9.8 m/s² until it reaches the floor 2.5 seconds later. A physics class is modeling the 1-D straight line motion as part of a class project They want construct a simple, but accurate, data table using the following equations: V₁+1 = V¡ + at and X₁+1 = X₁ + vi t which are familiar to you. Their instructor requires them to make the constant time interval At between rows in their table sufficiently small so that they have only a 4 percent error in the total distance traveled by the ball during the 2.5 seconds. What should be the value of At in their data table to cover the entire time interval and achieve the desired error?

A basketball is dropped from a walkway above Bartow Arena. It falls from rest with a constant acceleration of 9.8 m/s² until it reaches the floor 2.5 seconds later. A physics class is modeling the 1-D straight line motion as part of a class project They want construct a simple, but accurate, data table using the following equations: V₁+1 = V¡ + at and X₁+1 = X₁ + vi t which are familiar to you. Their instructor requires them to make the constant time interval At between rows in their table sufficiently small so that they have only a 4 percent error in the total distance traveled by the ball during the 2.5 seconds. What should be the value of At in their data table to cover the entire time interval and achieve the desired error?

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Distance and Speed

The speed, distance and time relation forms the basis of study of motion. You will need these concepts in the study of Mechanics. This concept is applicable for students who are studying the following courses –

Question

**Modeling 1-D Straight Line Motion - Educational Example**

A basketball is dropped from a walkway above Bartow Arena. It falls from rest with a constant acceleration of 9.8 m/s² until it reaches the floor 2.5 seconds later. A physics class is modeling this 1-D straight line motion as part of a class project. They want to construct a simple, but accurate, data table using the following equations:

\[ v_{i+1} = v_i + a \cdot t \]
\[ x_{i+1} = x_i + v_i \cdot t \]

Their instructor requires them to make the constant time interval \(\Delta t\) between rows in their table sufficiently small so that they have only a 4 percent error in the total distance traveled by the ball during the 2.5 seconds.

**Question:** What should be the value of \(\Delta t\) in their data table to cover the entire time interval and achieve the desired error?

### Explanation

#### Equations of Motion:
1. **Velocity Update:**
\[ v_{i+1} = v_i + a \cdot t \]
Where:
- \( v_{i+1} \) is the velocity at the next time step.
- \( v_i \) is the current velocity.
- \( a \) is the constant acceleration (9.8 m/s²).
- \( t \) is the time interval.

2. **Position Update:**
\[ x_{i+1} = x_i + v_i \cdot t \]
Where:
- \( x_{i+1} \) is the position at the next time step.
- \( x_i \) is the current position.
- \( v_i \) is the current velocity.
- \( t \) is the time interval.

#### Task:
To determine the appropriate time interval \(\Delta t\) so that the error in the total distance traveled by the basketball is within 4 percent over the 2.5-second fall.

By calculating and adjusting \(\Delta t\), we ensure the data table provides a precise representation of the basketball’s fall, crucial for the class project’s accuracy. This example showcases fundamental physics concepts like constant acceleration, kinematic equations, and the importance of precision in experimental physics.

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