Chapter 2: Problem 44
Calculate The position of a ball as a function of time is given by $$ x=3.0 \mathrm{~m}+(-5.0 \mathrm{~m} / \mathrm{s}) t $$ What is the position of the ball at \(1.5 \mathrm{~s}\) ?
Short Answer
Expert verified
The position of the ball at 1.5 seconds is -4.5 meters.
Step by step solution
01
Identify Given Equation
The position function of the ball is given as \( x = 3.0 \, \text{m} + (-5.0 \, \text{m/s}) t \). This equation shows that the ball starts at an initial position of \( 3.0 \text{ m} \) and moves at a velocity of \( -5.0 \text{ m/s} \).
02
Substitute the Time Value
We want to find the position of the ball at \( t = 1.5 \text{ s} \). Substitute \( t = 1.5 \) into the equation: \( x = 3.0 \, \text{m} + (-5.0 \, \text{m/s}) \times 1.5 \, \text{s} \).
03
Calculate the Product
Calculate the product of the velocity and time: \( -5.0 \, \text{m/s} \times 1.5 \, \text{s} = -7.5 \, \text{m} \).
04
Solve for Position
Substitute the calculated product into the equation: \( x = 3.0 \, \text{m} - 7.5 \, \text{m} \).
05
Calculate Final Result
Perform the subtraction to find the position: \( x = 3.0 - 7.5 = -4.5 \, \text{m} \). This means the ball is 4.5 meters in the negative direction from the origin after 1.5 seconds.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Position Function
In kinematics, the position function tells us where an object is located at a specific point in time. It is typically described in the form of an equation that relates an object's position to time. For example, in the exercise, the position of the ball is given by the equation:
- \( x = 3.0 \, \text{m} + (-5.0 \, \text{m/s}) \cdot t \)
Velocity
Velocity is a key component in understanding motion. It signifies the rate at which the object's position changes over time and includes both magnitude and direction. In our example, the velocity of the ball is:
- \(-5.0 \, \text{m/s}\)
Time
Time plays a crucial role in analyzing kinematic equations. In our problem, we are interested in the ball's position at a specific time, namely 1.5 seconds. By substituting \(t = 1.5 \, \text{s}\) into the position function, we can find the ball's exact position at that moment. Time helps us track how movements unfold and keeps our analysis grounded in real-world scenarios, allowing calculations that translate abstract equations into practical outcomes.
Displacement
Displacement refers to the change in position of an object, providing insight into the object’s relative motion from one point to another. In essence, it is the difference between the initial and final position:
- Initial position: \(3.0 \, \text{m}\)
- Final position: \(-4.5 \, \text{m}\)
Equation of Motion
The equation of motion is a mathematical expression that describes an object’s position related to time, taking into account constants such as velocity and initial position. In our scenario, the equation is straightforward:
- \( x = 3.0 \, \text{m} + (-5.0 \, \text{m/s}) \cdot t \)