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Ch. 02 - Describing Motion: Kinematics in One Dimension
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 02 - Describing Motion: Kinematics in One DimensionProblem 78a
Chapter 2, Problem 78a

The position of a ball rolling in a straight line is given by 𝓍 = 2.0 ― 3.6t + 1.7tΒ², where 𝓍 is in meters and t in seconds. What do the numbers 2.0, 3.6, and 1.7 refer to?

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The given equation for the position of the ball is 𝓍 = 2.0 - 3.6t + 1.7tΒ². This is a quadratic equation of motion, where the position 𝓍 is expressed as a function of time t.
The constant term 2.0 represents the initial position of the ball at t = 0. This is the position of the ball when the motion starts.
The coefficient -3.6 (associated with t) represents the initial velocity of the ball. It indicates the rate of change of position with respect to time at the start of the motion. The negative sign suggests the ball is moving in the opposite direction of the positive axis initially.
The coefficient 1.7 (associated with tΒ²) represents half of the acceleration of the ball. This term arises from the kinematic equation for uniformly accelerated motion: 𝓍 = 𝓍₀ + vβ‚€t + (1/2)atΒ². Therefore, the acceleration can be found by multiplying this coefficient by 2.
In summary, 2.0 is the initial position, -3.6 is the initial velocity, and 1.7 is half the acceleration of the ball.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Quadratic Function

The equation provided represents a quadratic function, which is a polynomial of degree two. In this context, the position of the ball as a function of time is expressed in the form 𝓍 = atΒ² + bt + c, where 'a', 'b', and 'c' are constants. Quadratic functions typically describe motion under uniform acceleration, making them essential for analyzing the ball's trajectory.

Coefficients in Motion Equations

In the equation 𝓍 = 2.0 - 3.6t + 1.7tΒ², the coefficients represent specific physical quantities. The constant term (2.0) indicates the initial position of the ball, while the linear term coefficient (-3.6) represents the initial velocity. The quadratic term coefficient (1.7) signifies the acceleration of the ball, which affects how its position changes over time.
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Kinematics

Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. It provides the framework for understanding how position, velocity, and acceleration are related. The equation given is a kinematic equation that allows us to predict the position of the ball at any time 't', based on its initial conditions and acceleration.
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