Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

CALC A remote-controlled car is moving in a vacant parking lot. The velocity of the car as a function of time is given by \(\hat{v} =[5.00 m/s - (0.0180 m/s^3)t^2] \hat{\imath} = [2.00 m/s + (0.550 m/s^2)t]\hat{j}\). (a) What are \(a_x(t)\) and \(a_y(t)\), the \(x\)- and \(y\)-components of the car's velocity as functions of time? (b) What are the magnitude and direction of the car's velocity at \(t\) = 8.00 s? (b) What are the magnitude and direction of the car's acceleration at \(t\) = 8.00 s?

Short Answer

Expert verified
(a) \( a_x(t) = -0.0360t \), \( a_y(t) = 0.550 \). (b) \( v = 6.98 \text{ m/s}, \) \( 66.8^\circ \). \( a = 0.620 \text{ m/s}^2 \), \( 117^\circ \).

Step by step solution

01

Identify the Velocity Components

From the given velocity vector \( \hat{v} \), we can identify the components:- The velocity in the \( x \)-direction: \( v_x = 5.00 \text{ m/s} - (0.0180 \text{ m/s}^3)t^2 \).- The velocity in the \( y \)-direction: \( v_y = 2.00 \text{ m/s} + (0.550 \text{ m/s}^2)t \).
02

Find the Acceleration Components

To find the acceleration components, differentiate the velocity components with respect to time:- For the \( x \)-component: \( a_x(t) = \frac{d}{dt} \left(5.00 - 0.0180t^2 \right) = -0.0360t \text{ m/s}^2 \).- For the \( y \)-component: \( a_y(t) = \frac{d}{dt} \left(2.00 + 0.550t \right) = 0.550 \text{ m/s}^2 \).
03

Evaluate the Velocity Components at t = 8.00 s

Substitute \( t = 8.00 \) s into the expressions for \( v_x \) and \( v_y \):- \( v_x(8) = 5.00 - (0.0180)(8)^2 = 2.85 \text{ m/s} \).- \( v_y(8) = 2.00 + (0.550)(8) = 6.40 \text{ m/s} \).
04

Calculate the Magnitude and Direction of Velocity

The magnitude of the velocity vector is given by \( v = \sqrt{v_x^2 + v_y^2} \):- \( v = \sqrt{(2.85)^2 + (6.40)^2} = 6.98 \text{ m/s} \).The direction \( \theta \) is the angle with respect to the \( x \)-axis:- \( \theta = \tan^{-1}\left( \frac{v_y}{v_x} \right) = \tan^{-1}\left( \frac{6.40}{2.85} \right) = 66.8^\circ \).
05

Evaluate the Acceleration Components at t = 8.00 s

Substitute \( t = 8.00 \) s into the expressions for \( a_x \) and \( a_y \):- \( a_x(8) = -0.0360 \times 8 = -0.288 \text{ m/s}^2 \).- \( a_y(8) = 0.550 \text{ m/s}^2 \).
06

Calculate the Magnitude and Direction of Acceleration

The magnitude of the acceleration vector is given by \( a = \sqrt{a_x^2 + a_y^2} \):- \( a = \sqrt{(-0.288)^2 + (0.550)^2} = 0.620 \text{ m/s}^2 \).The direction \( \phi \) is the angle with respect to the \( x \)-axis:- \( \phi = \tan^{-1}\left( \frac{a_y}{a_x} \right) = \tan^{-1}\left( \frac{0.550}{-0.288} \right) = 117^\circ \), meaning it is directed above the negative \( x \)-axis.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Velocity Components
Understanding the velocity components of an object is crucial in kinematics, as they help us determine how fast something is moving in a particular direction. In the case of the remote-controlled car, the velocity is given as a vector function of time:
\[\hat{v} =[5.00 \text{ m/s} - (0.0180 \text{ m/s}^3)t^2] \hat{\imath} + [2.00 \text{ m/s} + (0.550 \text{ m/s}^2)t]\hat{j}\]
Here are the components:
  • The velocity in the x-direction: \( v_x = 5.00 \text{ m/s} - (0.0180 \text{ m/s}^3)t^2 \)
  • The velocity in the y-direction: \( v_y = 2.00 \text{ m/s} + (0.550 \text{ m/s}^2)t \)
These equations tell us how the car's speed changes over time in the horizontal and vertical directions. At any time \( t \), you can substitute into these equations to find the speed of the car in the x and y directions. This is a foundational concept for solving problems in kinematics, as it allows for the decomposition of complex motion into simpler, perpendicular directional components.
Acceleration Components
The concept of acceleration is all about how the velocity of an object changes over time. In the context of the given exercise, the acceleration components can be found by differentiating the velocity components with respect to time.
For our remote-controlled car:
  • The acceleration in the x-direction is obtained by differentiating \( v_x \) with respect to time: \( a_x(t) = \frac{d}{dt}[5.00 - 0.0180t^2] = -0.0360t \text{ m/s}^2 \).
  • The acceleration in the y-direction is constant, since we differentiate \( v_y \): \( a_y(t) = \frac{d}{dt}[2.00 + 0.550t] = 0.550 \text{ m/s}^2 \).

Knowing these components, you can describe how the velocity is changing in both the horizontal and vertical directions. If you substitute a specific time into the equations for \( a_x \) and \( a_y \), you will discover how the object's velocity is accelerating at that moment. This is incredibly useful for predicting future motion and making real-time adjustments. Think of it as understanding the push and pull in each axis independently.
Magnitude and Direction
Magnitude and direction are essential when dealing with vector quantities like velocity and acceleration. They help us convert the vector into an understandable single value and its orientation in space.
To find the magnitude of velocity or acceleration, we use the formula:
  • For velocity: \( v = \sqrt{v_x^2 + v_y^2} \)
  • For acceleration: \( a = \sqrt{a_x^2 + a_y^2} \)
The direction is determined by the angle formed with the positive x-axis. For velocity, this is given by:
  • \( \theta = \tan^{-1}\left( \frac{v_y}{v_x} \right) \)
And for acceleration:
  • \( \phi = \tan^{-1}\left( \frac{a_y}{a_x} \right) \)

Let's apply this at \( t = 8.00 \text{ s} \):
  • Velocity magnitude: \( v = \sqrt{(2.85)^2 + (6.40)^2} = 6.98 \text{ m/s} \)
  • Direction with respect to the x-axis: \( \theta = 66.8^\circ \)
  • Acceleration magnitude: \( a = \sqrt{(-0.288)^2 + (0.550)^2} = 0.620 \text{ m/s}^2 \)
  • Direction with respect to the x-axis: \( \phi = 117^\circ \)

Thus, magnitude gives us a scalar context while direction provides the spatial insight. Together, they let us fully appreciate the behavior of the moving car in the parking lot. By mastering these concepts, you develop a deeper understanding of how vectors shape motion equations in physics.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

CALC A web page designer creates an animation in which a dot on a computer screen has position $$ \vec{r} =[34.0 cm +(2.5 cm/s^2)t^2] \hat{i} +(5.0 cm/s)t \hat{\jmath}.$$ (a) Find the magnitude and direction of the dot's average velocity between \(t\) = 0 and \(t\) = 2.0 s.(b) Find the magnitude and direction of the instantaneous velocity at \(t\) = 0, \(t\) = 1.0 s, and \(t\) = 2.0 s. (c) Sketch the dot's trajectory from \(t\) = 0 to \(t\) = 2.0 s, and show the velocities calculated in part (b).

A rock is thrown with a velocity \(v_0\), at an angle of \(\alpha_0\) from the horizontal, from the roof of a building of height \(h\). Ignore air resistance. Calculate the speed of the rock just before it strikes the ground, and show that this speed is independent of \(\alpha_0\).

A man stands on the roof of a 15.0-m-tall building and throws a rock with a speed of 30.0 m/s at an angle of 33.0\(^\circ\) above the horizontal. Ignore air resistance. Calculate (a) the maximum height above the roof that the rock reaches; (b) the speed of the rock just before it strikes the ground; and (c) the horizontal range from the base of the building to the point where the rock strikes the ground. (d) Draw \(x-t, y-t, v_x-t\), and \(v_y-t\) graphs for the motion.

A 2.7-kg ball is thrown upward with an initial speed of 20.0 m/s from the edge of a 45.0-m-high cliff. At the instant the ball is thrown, a woman starts running away from the base of the cliff with a constant speed of 6.00 m/s. The woman runs in a straight line on level ground. Ignore air resistance on the ball. (a) At what angle above the horizontal should the ball be thrown so that the runner will catch it just before it hits the ground, and how far does she run before she catches the ball? (b) Carefully sketch the ball's trajectory as viewed by (i) a person at rest on the ground and (ii) the runner.

At its Ames Research Center, NASA uses its large "20-G" centrifuge to test the effects of very large accelerations ("hypergravity") on test pilots and astronauts. In this device, an arm 8.84 m long rotates about one end in a horizontal plane, and an astronaut is strapped in at the other end. Suppose that he is aligned along the centrifuge's arm with his head at the outermost end. The maximum sustained acceleration to which humans are subjected in this device is typically 12.5\(g\). (a) How fast must the astronaut's head be moving to experience this maximum acceleration? (b) What is the \(difference\) between the acceleration of his head and feet if the astronaut is 2.00 m tall? (c) How fast in rpm (rev/min) is the arm turning to produce the maximum sustained acceleration?

See all solutions

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free