Question

A Honda Civic travels in a straight line along a road. The car's distance \(x\) from a stop sign is given as a function of time \(t\) by the equation \(x(t) = \alpha{t^2} - \beta{t^3}\), where \(\alpha =\) 1.50 m/s\(^2\) and \(\beta =\) 0.0500 m/s\(^3\). Calculate the average velocity of the car for each time interval: (a) \(t =\) 0 to \(t =\) 2.00 s; (b) \(t =\) 0 to \(t =\) 4.00 s; (c) \(t =\) 2.00 s to \(t =\) 4.00 s.

Short answer

(a) 2.8 m/s; (b) 5.2 m/s; (c) 7.6 m/s.

Step-by-step solution

Understand the Problem

The function given is \(x(t) = \alpha t^2 - \beta t^3\). We need to calculate the average velocity over the specified intervals. Average velocity, \( v_{avg} \), is defined as the total displacement divided by the total time taken, or \( v_{avg} = \frac{x(t_2) - x(t_1)}{t_2 - t_1}\).

Calculate x(t) for Each Time Interval (Part a)

For interval (a), \(t = 0\) to \(t = 2.00\) s, we calculate \(x(0)\) and \(x(2)\):- \(x(0) = \alpha (0)^2 - \beta (0)^3 = 0\)- \(x(2) = \alpha (2)^2 - \beta (2)^3 = 1.50 \times 4 - 0.0500 \times 8 = 6 - 0.4 = 5.6\) meters.

Calculate Average Velocity for Part a

Using the formula for average velocity, \(v_{avg} = \frac{x(2) - x(0)}{2 - 0}\):- \(v_{avg} = \frac{5.6 - 0}{2} = \frac{5.6}{2} = 2.8\) m/s.

Calculate x(t) for Each Time Interval (Part b)

For interval (b), \(t = 0\) to \(t = 4.00\) s, calculate \(x(0)\) and \(x(4)\):- \(x(0) = \alpha (0)^2 - \beta (0)^3 = 0\)- \(x(4) = \alpha (4)^2 - \beta (4)^3 = 1.50 \times 16 - 0.0500 \times 64 = 24 - 3.2 = 20.8\) meters.

Calculate Average Velocity for Part b

Using the formula for average velocity, \(v_{avg} = \frac{x(4) - x(0)}{4 - 0}\):- \(v_{avg} = \frac{20.8 - 0}{4} = \frac{20.8}{4} = 5.2\) m/s.

Calculate x(t) for Each Time Interval (Part c)

For interval (c), \(t = 2.00\) to \(t = 4.00\) s, calculate \(x(2)\) and \(x(4)\) (computed earlier):- \(x(2) = 5.6\) meters (already calculated in Part a)- \(x(4) = 20.8\) meters (already calculated in Part b).

Calculate Average Velocity for Part c

Using the formula for average velocity, \(v_{avg} = \frac{x(4) - x(2)}{4 - 2}\):- \(v_{avg} = \frac{20.8 - 5.6}{2} = \frac{15.2}{2} = 7.6\) m/s.

Key concepts

Kinematics

Kinematics is the branch of physics that deals with the motion of objects without considering the forces that cause the motion. It is a key concept in understanding how objects move from one place to another. When studying kinematics, we focus on several key variables:

• Displacement - The change in position of an object.
• Velocity - The rate of change of displacement with respect to time.
• Acceleration - The rate of change of velocity over time.

Kinematics uses these concepts to predict the future position and velocity of an object. In this exercise, we analyze the Honda Civic's motion by examining how its position changes over time using the given displacement equation.

Distance-Time Relation

The distance-time relation is foundational in kinematics. It helps us understand how the distance traveled by an object is related to the time elapsed. For our exercise involving the Honda Civic, this relationship is represented by the equation:\[x(t) = \alpha t^2 - \beta t^3\]where the distance, or position, \( x(t) \) changes as time \( t \) progresses.

• \( \alpha t^2 \) reflects the initial acceleration-driven change in position.
• \( \beta t^3 \) depicts how the impact of acceleration diminishes over time due to the cubed time term.

By plugging in different values for \( t \), we can see how the position of the car changes at various times, forming a clear picture of the motion.

Honda Civic Motion

In this exercise, we're observing the motion of a Honda Civic as it travels in a straight line. The movement of the car is analyzed through the provided displacement equation. Let's break down the car's motion:- From rest, at \( t = 0 \), the car is at the origin.- As time passes (\( t = 2 \) seconds and \( t = 4 \) seconds), we calculate the car’s position using the equation and find how far it travels during these intervals.This simple yet powerful analysis allows us to understand its average velocity for any given time period. The average velocity tells us the uniform speed needed to cover the same distance in the same time frame.

Displacement Formula

The displacement formula used in our exercise is a mathematical representation of the motion of the Honda Civic. It's given by:\[x(t) = \alpha t^2 - \beta t^3\]In this formula:

• \( \alpha \) represents the initial acceleration, measured in meters per second squared (m/s\(^2\)).
• \( \beta \) denotes the rate at which acceleration impacts the motion over time, measured in meters per second cubed (m/s\(^3\)).

The displacement equation helps us determine the car's exact position at any time \( t \). By understanding displacement, we can calculate average velocity, which is crucial for determining how quickly the car is moving from one point to another. Average velocity is calculated as:\[v_{avg} = \frac{x(t_2) - x(t_1)}{t_2 - t_1}\]This gives us a practical measure of the car’s motion over a designated time period.