Question

A rabbit runs in a garden such that the \(x\) - and \(y\) -components of its displacement as functions of time are given by \(x(t)=-0.45 t^{2}-6.5 t+25\) and \(y(t)=0.35 t^{2}+8.3 t+34 .(\) Both \(x\) and \(y\) are in meters and \(t\) is in seconds.) a) Calculate the rabbit's position (magnitude and direction) at \(t=10.0 \mathrm{~s}\). b) Calculate the rabbit's velocity at \(t=10.0 \mathrm{~s}\). c) Determine the acceleration vector at \(t=10.0 \mathrm{~s}\).

Short answer

Answer: To find the magnitude and angles of the position, velocity, and acceleration vectors at t=10.0s, follow these steps: 1. Calculate the rabbit's position at t=10.0s by substituting the given time into the x(t) and y(t) functions, then find the magnitude (R) and angle (θ) of the position vector using trigonometry. 2. Calculate the rabbit's velocity at t=10.0s by taking the first derivative of the x(t) and y(t) functions and substituting the given time, then find the magnitude (V) and angle (φ) of the velocity vector using trigonometry. 3. Determine the acceleration vector at t=10.0s by taking the second derivatives of both x(t) and y(t), as they are constant and do not depend on time.

Step-by-step solution

a) Calculate rabbit's position at t=10.0s

First, we need to find the x and y components of the rabbit's position at t=10.0s. To do that, we'll plug in t=10.0s into the given x(t) and y(t) functions: $$x(10) = -0.45(10)^2 - 6.5(10) + 25$$ $$y(10) = 0.35(10)^2 + 8.3(10) + 34$$

Calculating the magnitude and direction of the position vector

Next, we'll calculate the magnitude of the position vector by using the Pythagorean theorem: $$R =\sqrt{x(10)^2 + y(10)^2}$$ To find the direction of the position vector, we'll calculate its angle (θ) from the positive x-axis using the inverse tangent function: $$θ = \tan^{-1} \left(\frac{y(10)}{x(10)}\right)$$

b) Calculate the rabbit's velocity at t=10.0s

To find the velocity components, we'll take the first derivative of both x(t) and y(t): $$\frac{dx(t)}{dt} = -0.9t - 6.5$$ $$\frac{dy(t)}{dt} = 0.7t + 8.3$$ Then we'll plug in t=10.0s to find the velocity components at that time: $$V_x = -0.9(10) - 6.5$$ $$V_y = 0.7(10) + 8.3$$

Calculating the magnitude and direction of the velocity vector

Now, we'll calculate the magnitude of the velocity vector using the Pythagorean theorem: $$V = \sqrt{V_x^2 + V_y^2}$$ To find the direction of the velocity vector, we'll calculate its angle (φ) from the positive x-axis using the inverse tangent function: $$φ = \tan^{-1} \left(\frac{V_y}{V_x}\right)$$

c) Determine the acceleration vector at t=10.0s

Finally, to find the acceleration components, we'll take the second derivative of both x(t) and y(t): $$\frac{d^2x(t)}{dt^2} = -0.9$$ $$\frac{d^2y(t)}{dt^2} = 0.7$$ Since the second derivatives are constants, the acceleration does not depend on time. Therefore, the acceleration vector at t=10.0s (and at any other time) is: $$A_x = -0.9 \ \mathrm{m/s^2}$$ $$A_y = 0.7 \ \mathrm{m/s^2}$$

Key concepts

Position Vector

The position vector is a fundamental concept in kinematics, representing the location of an object in space relative to a chosen origin. In the context of the rabbit's movement, this vector indicates where the rabbit is in the garden at a specific moment.

Application in Our Exercise:Following the exercise, we calculate the rabbit's position vector at a particular time, t=10 seconds, by evaluating the given functions for x(t) and y(t). The calculated x and y components then form a vector \(\textbf{r}(t) = x(t)\textbf{i} + y(t)\textbf{j}\), where \textbf{i} and \textbf{j} are the unit vectors in the horizontal and vertical directions of the garden, respectively.

• Representing Directions: To visualize it, imagine a grid laid over the garden with the starting point as the origin. The rabbit's position can be pinpointed by moving along the x-axis by the x-component and then parallel to the y-axis by the y-component.
• Magnitude: The magnitude of this vector is the length of the straight line from the origin to the rabbit's location, which we find using the Pythagorean theorem, emphasizing the distance the rabbit has covered.
• Direction: The direction is then conceptualized by the angle the vector makes with the x-axis, which is obtained using the inverse tangent function and provides us with an orientation in the two-dimensional space of the garden.

Velocity Vector

In kinematic terms, the velocity vector defines not just how fast, but in which direction an object is moving. It's a vector that quantifies an object's speed and the direction of that speed at a given instant.

Calculating Velocity in the Exercise:We determine the rabbit's velocity vector by differentiating the position functions with respect to time. This process gives us the velocity components in the x and y directions, labeled V_x and V_y. At time t=10 seconds, these components reveal how rapidly and towards which direction (positive or negative axis direction) the rabbit is moving right at that snapshot in time.

• Magnitude of Velocity: To find the overall speed, we use the magnitude of the velocity vector, calculated similarly to the position vector, with the Pythagorean theorem, thus synthesizing both the horizontal and vertical components into a singular term representing total speed.
• Velocity Direction: The direction of motion is expressed by the angle the velocity vector makes with the positive x-axis, informing us about the trajectory of the rabbit's journey through the garden at that instant.

Velocity helps us understand not merely how position changes, but in a more refined manner - articulating the swiftness and directionality of that change.

Acceleration Vector

Acceleration is a vector quantity that represents the rate at which an object's velocity vector changes with time. It tells us about the object's speeding up, slowing down, and altering its course.

Acceleration in Our Rabbit Scenario:The acceleration vector for the rabbit's motion is determined by taking the second derivative of the position functions regarding time. This gives us the acceleration components A_x and A_y, which are constant values in this case, indicating that the rabbit's velocity changes at a steady rate throughout its romp in the garden.

• Constant Acceleration: Because the acceleration components are independent of time, we deduce that the rabbit experiences a consistent force influencing its movement, possibly due to the garden's landscape or its own muscular exertion.
• Acceleration Components: The x-component A_x and y-component A_y tell us about the changes in velocity along the x-axis and y-axis, respectively, which when combined, describe how the rabbit is moving: slowing down, speeding up, and/or turning.

When students visualize the acceleration vector's constants, it aids in understanding how motion can be influenced by persistent forces, like gravity on a dropping ball, or the garden's terrain affecting the rabbit.