Question

A golf cart moves with a velocity of \(8 \mathrm{~m} / \mathrm{s}\). Is the displacement of the golf cart from \(t=0\) to \(t=5 \mathrm{~s}\) greater than, less than, or equal to its displacement from \(t=5 \mathrm{~s}\) to \(t=10 \mathrm{~s}\) ? Explain.

Short answer

The displacements are equal in both time intervals.

Step-by-step solution

Understand the Problem

We need to determine if the displacement of a golf cart during two time intervals, each 5 seconds long, is greater than, less than, or equal. The golf cart moves at a constant velocity of \(8 \, \text{m/s}\).

Recall the Formula for Displacement

Displacement \(s\) can be calculated using the formula:\[ s = v \times t \]where \(v\) is the velocity and \(t\) is the time.

Calculate Displacement from \(t=0\) to \(t=5\)

For the first time interval, the time elapsed \(t=5\,\text{s} - 0\,\text{s} = 5\,\text{s}\). Using the displacement formula:\[ s_1 = 8 \, \text{m/s} \times 5 \, \text{s} = 40 \, \text{m} \]

Calculate Displacement from \(t=5\) to \(t=10\)

For the second time interval, the time elapsed \(t=10\,\text{s} - 5\,\text{s} = 5\,\text{s}\). Using the displacement formula:\[ s_2 = 8 \, \text{m/s} \times 5 \, \text{s} = 40 \, \text{m} \]

Compare the Displacements

Both displacements \(s_1\) and \(s_2\) are equal, as both equal \(40 \, \text{m}\). Therefore, the displacements in the two intervals are equal.

Key concepts

Displacement

Displacement is a fundamental concept in kinematics and refers to the change in position of an object. It's important to note that displacement accounts for direction as well as distance. This differentiates it from the total path length traveled, known as distance, which does not consider direction.

To calculate displacement, we use the formula:

• \( s = v \times t \)

Where:

• \( s \) is the displacement
• \( v \) is the velocity
• \( t \) is the time

This equation provides a straight-line connection between the starting and ending points of an object's motion, at a constant velocity. Understanding displacement is crucial for solving problems in uniform motion, as it tells us how far and in which direction an object has moved from a starting point to an endpoint.

Constant Velocity

Constant velocity means that an object is moving at a steady speed in a straight line, without speeding up or slowing down. When analyzing objects in uniform motion, constant velocity simplifies calculations because the object covers the same amount of distance in each equal time interval.

• In the exercise, the golf cart's constant velocity is \(8 \, \text{m/s}\), meaning it travels 8 meters every second.

This concept simplifies analysis because we can focus on multiplying the velocity by time to get displacement, without worrying about changes in speed.
In practical situations, constant velocity is often an idealization. However, understanding this concept forms the basis for more complex motion analysis.

Kinematics

Kinematics is the study of motion without considering the forces that cause it. It's the foundation for analyzing how objects move and is critical for understanding concepts such as displacement and velocity.

Kinematics involves:

• Describing the motion of objects using mathematics
• Using variables like displacement, velocity, and time to analyze motion

In our problem, kinematics is used to determine that the golf cart, moving at a constant velocity, has equal displacements in two equal time intervals. This exercise illustrates the principle that, in the absence of acceleration, displacement can be linearly predicted using straightforward multiplication of velocity and time.
The fair simplicity of kinematic equations makes them versatile tools in both academic exercises and real-world applications, such as in engineering and physics.