Question

The position of a race car on a straight track is given as \(x=a t^{3}+b t^{2}+c,\) where \(a=2.0 \mathrm{~m} / \mathrm{s}^{3}, b=2.0 \mathrm{~m} / \mathrm{s}^{2}\), and \(c=3.0 \mathrm{~m}\). a) What is the car's position between \(t=4.0 \mathrm{~s}\) and \(t=9.0 \mathrm{~s}\) ? b) What is the average speed between \(t=4.0 \mathrm{~s}\) and \(t=9.0 \mathrm{~s} ?\)

Short answer

Part A: For the given position function \(x = at^3 + bt^2 + c\), at \(t = 4.0\:s\), the car's position is \(171\:m\), and at \(t = 9.0\:s\), the position is \(1605\:m\). Part B: The average speed between \(t = 4.0\:s\) and \(t = 9.0\:s\) is \(286.8\:m/s\).

Step-by-step solution

Understanding the problem and given information

We are given the position function: $$x = at^3 + bt^2 + c$$ with given values for a, b and c as: \(a = 2.0\:m/s^3, b = 2.0\:m/s^2\), and \(c = 3.0\:m\). Our task in part (a) is to determine the car's position at \(t = 4.0\:s\) and \(t = 9.0\:s\).

Calculate the position at the given times

Plug in the values for a, b, c and the given time values into the position function: For \(t = 4.0\:s\): $$x = 2(4)^3 + 2(4)^2 + 3$$ For \(t = 9.0\:s\): $$x = 2(9)^3 + 2(9)^2 + 3$$ Now, we just need to compute the position which will be our answer for part (a).

Answers for part a

After the calculations in the previous step, we get: For \(t = 4.0\:s\) , the position is \(171\:m\) For \(t = 9.0\:s\), the position is \(1605\:m\) The positions of the car between \(t = 4.0 \mathrm{~s}\) and \(t=9.0 \mathrm{~s}\) are \(171\:m\) and \(1605\:m\). This is our answer for part (a).

Understand how to calculate average speed

The average speed is calculated by dividing the total distance traveled by the total time it takes to travel that distance. In this case, the time interval between the two positions is from \(4\:s\) to \(9\:s\), so the time interval is \(9\:s - 4\:s = 5\:s\). The position at \(t=4\:s\) is \(171\:m\) and at \(t=9\:s\) is \(1605\:m\). So the change in position is \(1605\:m - 171m = 1434\:m\).

Calculate the average speed

Now we can calculate the average speed during the interval \(t = 4.0 \mathrm{~s}\) to \(t =9.0 \mathrm{~s}\) as: $$v_{\text{average}} = \frac{\Delta x}{\Delta t} = \frac{1434\:m}{5\:s}$$

Answer for part b

After calculations in the previous step, we get \(v_{\text{average}} = 286.8\:m/s\). This is our answer for part (b).

Key concepts

Motion in Physics

In physics, motion is defined as the change in position of an object over time. Motion is a fundamental concept that applies to everything from celestial bodies orbiting in space to race cars speeding down a track, as seen in our exercise. To describe motion, we often refer to an object's displacement, speed, velocity, and acceleration.

Displacement is the shortest distance from the initial to the final position of an object and is a vector quantity, meaning it has both magnitude and direction. The speed of an object is the rate at which it covers distance, but without a specified direction, it's a scalar quantity. On the other hand, velocity includes direction and is therefore a vector quantity.

Understanding motion involves analyzing various kinematic scenarios, like the motion of our race car on a straight track, which can be predicted and calculated using formulas and position-time relationships.

Position-Time Relationship

The relationship between position and time is crucial when studying the motion of objects. It tells us how an object's position changes over time. In our exercise, we're given a mathematical expression that relates the position of the race car to time, known as the position function. It is a basic example of how motion can be represented mathematically. The position function can vary greatly, from simple linear relationships to more complex polynomial equations, depending on the nature of the motion.

In kinematics, the position-time graph is a valuable tool for visualizing motion. A straight diagonal line on such a graph would indicate constant speed, while a curved line suggests changing speed - acceleration or deceleration. Understanding this concept helps in dissecting complex motion and is foundational for solving kinematic problems, like determining the average speed of a race car between two points in time.

Kinematic Equations

Kinematic equations, also known as the equations of motion, are used to describe the motion of objects without considering the forces that cause the motion. These equations relate the variables of displacement, initial velocity, final velocity, acceleration, and time.

The most basic kinematic equation is one that equates distance traveled with the average speed multiplied by time, which looks like this in its simplest form: \[ d = vt \] However, the actual motion of objects can be more complex, and may require more sophisticated equations involving powers of time, as evident in our exercise with the race car's position function. Understanding and applying these kinematic equations allows us to predict the future positions and velocities of moving objects, making it a key concept in physics, engineering, and various other fields requiring motion analysis.