/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 225 At the instant shown, car \(A\) ... [FREE SOLUTION] | 91影视

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Chapter 12: Problem 225

At the instant shown, car \(A\) has a speed of \(20 \mathrm{~km} / \mathrm{h},\) which is being increased at the rate of \(300 \mathrm{~km} / \mathrm{h}^{2}\) as the car enters the expressway. At the same instant, \(\operatorname{car} B\) is decelerating at \(250 \mathrm{~km} / \mathrm{h}^{2}\) while traveling forward at \(100 \mathrm{~km} / \mathrm{h} .\) Determine the velocity and acceleration of \(A\) with respect to \(B\).

Short Answer

Expert verified
The velocity of car A with respect to car B is -22.2222 m/s in the direction opposite to the path, and its acceleration is 0.042438 m/s^2 in the direction of the path.

Step by step solution

01

Convert the given data into compatible units

The given speeds are in km/h, whereas the acceleration rates are given in km/h^2. To make the computations easier, convert these units into m/s and m/s^2 respectively. 1 km/h = 0.277778 m/s and \(1 \, \text{km/h}^2 = 0.00007716 \, \text{m/s}^2\). Therefore, the velocity of A is \(5.5556 \, \text{m/s}\) and its acceleration is \(0.023148 \, \text{m/s}^2\), and the velocity of B is \(27.7778 \, \text{m/s}\) and its acceleration is \(-0.01929 \, \text{m/s}^2\). This negative sign indicates that car B is decelerating.
02

Calculate the relative velocity and acceleration

The relative velocity and acceleration can be calculated using the formulas v_{AB} = v_A - v_B and a_{AB} = a_A - a_B respectively. Substituting the values from step 1, we find that v_{AB} = \(5.5556 \, \text{m/s} - 27.7778 \, \text{m/s} = -22.2222 \, \text{m/s}\), and a_{AB} = \(0.023148 \, \text{m/s}^2 - (-0.01929 \, \text{m/s}^2) = 0.042438 \, \text{m/s}^2\). The negative sign of v_{AB} indicates that the relative velocity of A with respect to B is in the opposite direction of the path, and a_{AB} indicates that A is accelerating relative to B.

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Key Concepts

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

Velocity Conversion
Grasping the concept of velocity conversion is crucial when dealing with mechanics problems. Velocity describes the speed of an object along with its direction of movement and is typically measured in meters per second (m/s) or kilometers per hour (km/h). Conversion between these units involves a straightforward mathematical relationship: 1 km/h is equivalent to approximately 0.277778 m/s.

To convert km/h to m/s, you multiply the speed by this factor. Conversely, to convert m/s to km/h, you divide by the same factor. For instance, a car traveling at 20 km/h is moving at about 5.556 m/s鈥攁 piece of useful information for computing relative velocities or kinematic equations in a consistent unit system.
Acceleration Conversion
Just as with velocity, acceleration conversion is vital because acceleration expresses the rate of change of velocity and typically comes in units like meters per second squared (m/s2) or kilometers per hour squared (km/h2).

To convert from km/h2 to m/s2, a similar conversion factor is used: 1 km/h2 equals approximately 0.00007716 m/s2. This means that if a car's speed is increasing at a rate of 300 km/h2, it is actually accelerating at 0.023148 m/s2 when expressed in standard SI units, which are essential for accurate calculations in physics.
Relative Motion Analysis
The concept of relative motion analysis allows us to understand the movement of objects with respect to one another. When studying relative motion, it is important to choose a reference frame. For example, if you're sitting in a moving car A observing another car B, car B's velocity and acceleration relative to you can differ from its absolute measures.

In our exercise, we used the formulae vAB = vA - vB for relative velocity and aAB = aA - aB for relative acceleration. The signs of the resulting values can indicate the direction of movement or acceleration relative to the chosen reference point, which is critical for understanding interactions between moving objects.
Deceleration
The term deceleration refers to a decrease in the speed of an object, or negative acceleration. In our context, it is represented by a negative acceleration value. For example, a car decelerating at 250 km/h2 (converted to -0.01929 m/s2) is slowing down. Deceleration is critical in safety considerations, such as determining stopping distances for vehicles. It's important to distinguish between speed and velocity here; deceleration affects the magnitude of velocity (speed), but it's the sign of acceleration that tells us if the object is slowing down or speeding up relative to the chosen direction.

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Most popular questions from this chapter

A particle moves along a straight line with an acceleration of \(a=5 /\left(3 s^{1 / 3}+s^{5 / 2}\right) \mathrm{m} / \mathrm{s}^{2},\) where \(s\) is in meters. Determine the particle's velocity when \(s=2 \mathrm{~m}\), if it starts from rest when \(s=1 \mathrm{~m}\). Use a numerical method to evaluate the integral.

A passenger in an automobile observes that raindrops make an angle of \(30^{\circ}\) with the horizontal as the auto travels forward with a speed of \(60 \mathrm{~km} / \mathrm{h}\). Compute the terminal (constant) velocity \(\mathbf{v}_{r}\) of the rain if it is assumed to fall vertically.

At the instant \(\theta=30^{\circ}\), the cam rotates with a clockwise angular velocity of \(\dot{\theta}=5 \mathrm{rad} / \mathrm{s}\) and, angular acceleration of \(\ddot{\theta}=6 \mathrm{rad} / \mathrm{s}^{2} .\) Determine the magnitudes of the velocity and acceleration of the follower \(\operatorname{rod} A B\) at this instant. The surface of the cam has a shape of a lima莽on defined by \(r=(200+100 \cos \theta) \mathrm{mm}\)

12-221. A car is traveling north along a straight road at \(50 \mathrm{~km} / \mathrm{h}\). An instrument in the car indicates that the wind is coming from the east. If the car's speed is \(80 \mathrm{~km} / \mathrm{h},\) the instrument indicates that the wind is coming from the northeast. Determine the speed and direction of the wind.

The particle travels along the path defined by the parabola \(y=0.5 x^{2}\). If the component of velocity along the \(x\) axis is \(v_{x}=(5 t) \mathrm{m} / \mathrm{s}\), where \(t\) is in seconds, determine the particle's distance from the origin \(O\) and the magnitude of its acceleration when \(t=1 \mathrm{~s}\). When \(t=0, x=0, y=0\).
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