Question

The velocity as a function of time for a car on an amusement park ride is given as \(v=A t^{2}+B t\) with constants \(A=2.0 \mathrm{~m} / \mathrm{s}^{3}\) and \(B=1.0 \mathrm{~m} / \mathrm{s}^{2} .\) If the car starts at the origin, what is its position at \(t=3.0 \mathrm{~s} ?\)

Short answer

Answer: The position of the car at t=3.0s is \(22.5 \mathrm{~m}\) from the origin.

Step-by-step solution

Write down the given information

We have the velocity function \(v(t) = 2.0t^2 + 1.0t\), with given constants \(A=2.0 \mathrm{~m} / \mathrm{s}^{3}\) and \(B=1.0 \mathrm{~m} / \mathrm{s}^{2}\).

Integrate the velocity function to get the displacement function

We can find the displacement function by integrating the velocity function with respect to time t: \(x(t) = \int v(t) dt = \int (2.0t^2 + 1.0t) dt\)

Find the indefinite integral of the function

Now, let's integrate the function: \(x(t) = \int (2.0t^2) dt + \int (1.0t) dt\) Using the power rule, we have: \(x(t) = 2.0 \int t^2 dt + 1.0\int t dt = 2.0 (\frac{1}{3}t^3) + 1.0(\frac{1}{2}t^2) + C\) Now, our position function looks like: \(x(t) = \frac{2}{3}t^3 + \frac{1}{2}t^2 + C\)

Determine the constant of integration

Given that the car starts at the origin, we know that the initial position at \(t=0\) is \(x(0) = 0\). Plugging this into the position function, we can find the constant C: \(0 = \frac{2}{3}(0)^3 + \frac{1}{2}(0)^2 + C\) \(C=0\) So, the position function becomes: \(x(t)=\frac{2}{3}t^3 + \frac{1}{2}t^2\)

Find the position at \(t=3.0s\)

Now, plug in the given time \(t=3.0s\) into our position function: \(x(3.0) = \frac{2}{3}(3.0)^3 + \frac{1}{2}(3.0)^2\) \(x(3.0) = \frac{2}{3}(27) + \frac{1}{2}(9) = 18 + 4.5 = 22.5\) So, at \(t=3.0s\), the position of the car is \(22.5 \mathrm{~m}\) from the origin.

Key concepts

Velocity-Time Relationship

Understanding the relationship between velocity and time is crucial in kinematics, the branch of physics that deals with motion. Velocity is the rate of change of position with respect to time and is a vector quantity, meaning it has both magnitude and direction. The relationship can be expressed mathematically as a velocity-time function, often denoted as v(t), which describes how an object's velocity changes over time.

In our example, the velocity-time function is given by the equation v(t) = 2.0t^2 + 1.0t. At any specific time, t, this equation allows us to calculate the velocity of the car on the amusement park ride. By analyzing this function, we can gain insights into the car's acceleration and ultimately determine its position at any given moment.

Integration in Physics

Integration is a mathematical operation that combines small pieces to determine the total or whole. In physics, and particularly kinematics, integration enables us to derive quantities that change cumulatively over time, such as displacement from a velocity-time function. It is the reverse process of differentiation.

In the context of our exercise, the car's displacement, which is the total distance moved in a particular direction, is found by integrating its velocity function over time. This process turns the rate of motion (velocity) back into the total amount of motion (displacement), a concept critical for understanding motion in a given scenario. By integrating the velocity function v(t) from zero to a particular time t, we obtain the object's net displacement during that interval.

Displacement Calculation

Displacement represents the change in position of an object. It's a vector quantity that accounts for both magnitude and direction. To compute displacement, we integrate the velocity over the time interval of interest.

In our exercise, the integration of the velocity-time relationship v(t) gave us the formula for displacement x(t). With the initial condition that the car starts at the origin, the constant of integration is determined to be zero, simplifying our calculation. Therefore, we can easily calculate the displacement at any time t by just plugging the time value into our displacement function.

Equations of Motion

Equations of motion are a set of formulas that describe the relationship between velocity, displacement, acceleration, and time. They are derived from the basic principles of kinematics and Newton's laws of motion.

The example we have been considering makes use of one such equation of motion, obtained through the integration of a velocity-time function. These equations enable us to predict the future position and velocity of an object undergoing uniform acceleration. In the absence of acceleration, these equations simplify further, representing uniform motion. It's important to recognize that the existence of constants like A and B in the given velocity function, and their respective units, gives us insight into the object's acceleration (A) and initial velocity (B) when t=0.