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Chapter 3: Problem 1

A squirrel has \(x\)- and \(y\)-coordinates (1.1 m, 3.4 m) at time \(t_1\) = 0 and coordinates (5.3 m, -0.5 m) at time \(t_2\) = 3.0 s. For this time interval, find (a) the components of the average velocity, and (b) the magnitude and direction of the average velocity.

Short Answer

Expert verified
(a) \(v_{x_{avg}} = 1.4\) m/s, \(v_{y_{avg}} = -1.3\) m/s; (b) Magnitude: 1.91 m/s, Direction: -42.6掳.

Step by step solution

01

Find the Components of Displacement

Determine the displacement components in the x- and y-directions. The displacement in the x-direction is the final x-coordinate minus the initial x-coordinate:\[\Delta x = x_2 - x_1 = 5.3 \text{ m} - 1.1 \text{ m} = 4.2 \text{ m}\]Similarly, the displacement in the y-direction is:\[\Delta y = y_2 - y_1 = -0.5 \text{ m} - 3.4 \text{ m} = -3.9 \text{ m}\]
02

Calculate the Components of Average Velocity

The average velocity components can be found by dividing the displacement components by the time interval. For the x-direction:\[v_{x_{avg}} = \frac{\Delta x}{\Delta t} = \frac{4.2 \text{ m}}{3.0 \text{ s}} = 1.4 \text{ m/s}\]And for the y-direction:\[v_{y_{avg}} = \frac{\Delta y}{\Delta t} = \frac{-3.9 \text{ m}}{3.0 \text{ s}} = -1.3 \text{ m/s}\]
03

Compute the Magnitude of Average Velocity

The magnitude of the average velocity is calculated using the Pythagorean theorem combining the x and y components:\[v_{avg} = \sqrt{v_{x_{avg}}^2 + v_{y_{avg}}^2} = \sqrt{(1.4 \text{ m/s})^2 + (-1.3 \text{ m/s})^2} = \sqrt{1.96 + 1.69} = \sqrt{3.65} \approx 1.91 \text{ m/s}\]
04

Determine the Direction of Average Velocity

The direction of the average velocity is given by the angle \(\theta\) with respect to the positive x-axis, using the arctangent function,\[\theta = \tan^{-1}\left(\frac{v_{y_{avg}}}{v_{x_{avg}}}\right) = \tan^{-1}\left(\frac{-1.3}{1.4}\right)\]Calculating this gives approximately \(\theta \approx -42.6^\circ\). The angle is measured clockwise from the positive x-axis.

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Key Concepts

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

Displacement
Displacement refers to the change in position of an object. It is a vector quantity, meaning it has both magnitude and direction. Here, displacement is calculated individually for each coordinate, and together they describe the overall change in position. To find the displacement for each component, you subtract the initial position from the final position:
  • For the x-direction: \[\Delta x = x_2 - x_1 = 5.3 \, \text{m} - 1.1 \, \text{m} = 4.2 \, \text{m}\]
  • For the y-direction: \[\Delta y = y_2 - y_1 = -0.5 \, \text{m} - 3.4 \, \text{m} = -3.9 \, \text{m}\]
The x-component of displacement indicates movement towards the right, while the negative y-component indicates movement downwards. The combination of these components gives the squirrel's overall displacement over the given time interval.
Velocity Components
Velocity components are crucial for understanding how an object's speed is distributed along different directions. For this problem, the velocity components represent how far the squirrel moves per second in the horizontal and vertical directions. They are found by dividing the displacement components by the time over which the motion occurs:
  • X-component of average velocity: \[v_{x_{avg}} = \frac{\Delta x}{\Delta t} = \frac{4.2 \, \text{m}}{3.0 \, \text{s}} = 1.4 \, \text{m/s}\]
  • Y-component of average velocity: \[v_{y_{avg}} = \frac{\Delta y}{\Delta t} = \frac{-3.9 \, \text{m}}{3.0 \, \text{s}} = -1.3 \, \text{m/s}\]
From these calculations, the squirrel's average horizontal velocity is 1.4 m/s to the right, and average vertical velocity is -1.3 m/s downward. This breakdown helps in visualizing how the total velocity is oriented in space.
Magnitude of Velocity
The magnitude of velocity gives an overall measure of speed, regardless of direction. It is a scalar quantity, calculated by combining the velocity components using the Pythagorean theorem. This gives the speed at which the squirrel is moving through space:\[v_{avg} = \sqrt{v_{x_{avg}}^2 + v_{y_{avg}}^2} = \sqrt{(1.4 \, \text{m/s})^2 + (-1.3 \, \text{m/s})^2}\]Calculating further, we have:\[v_{avg} = \sqrt{1.96 + 1.69} = \sqrt{3.65} \approx 1.91 \, \text{m/s}\]This value, 1.91 m/s, represents the actual speed of the squirrel over the time interval, regardless of the path it took from start to finish.
Direction of Velocity
The direction of velocity is expressed as an angle, giving the precise orientation of the velocity vector with respect to a reference line, usually the positive x-axis. It is calculated using the ratio of the velocity components:\[\theta = \tan^{-1}\left(\frac{v_{y_{avg}}}{v_{x_{avg}}}\right) = \tan^{-1}\left(\frac{-1.3}{1.4}\right)\]Solving for the angle gives:\[\theta \approx -42.6^\circ\]The negative angle indicates a direction clockwise from the positive x-axis. This helps visualize that the squirrel's movement was not straight but tilted downward as it moved horizontally.

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Most popular questions from this chapter

A 124-kg balloon carrying a 22-kg basket is descending with a constant downward velocity of 20.0 m/s. A 1.0-kg stone is thrown from the basket with an initial velocity of 15.0 m/s perpendicular to the path of the descending balloon, as measured relative to a person at rest in the basket. That person sees the stone hit the ground 5.00 s after it was thrown. Assume that the balloon continues its downward descent with the same constant speed of 20.0 m/s. (a) How high is the balloon when the rock is thrown? (b) How high is the balloon when the rock hits the ground? (c) At the instant the rock hits the ground, how far is it from the basket? (d) Just before the rock hits the ground, find its horizontal and vertical velocity components as measured by an observer (i) at rest in the basket and (ii) at rest on the ground.

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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?

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