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Chapter 12: Problem 75

A particle travels along the curve from \(A\) to \(B\) in \(2 \mathrm{~s}\). It takes \(4 \mathrm{~s}\) for it to go from \(B\) to \(C\) and then \(3 \mathrm{~s}\) to go from \(C\) to \(D\). Determine its average speed when it goes from \(A\) to \(D\).

Short Answer

Expert verified
The average speed can be calculated as distance travelled 'd' (unknown) divided by total time taken i.e., 9 seconds.

Step by step solution

01

Identify the given information

We know that the time taken to travel from A to B is 2 seconds, from B to C is 4 seconds and from C to D is 3 seconds.
02

Sum the times for each segment

To find the total time of travel, we need to add all of these individual times together: \(t_{total} = t_{AB} + t_{BC} + t_{CD} = 2s + 4s + 3s = 9s \)
03

Determine the average speed

Average speed is total distance divided by total time. We may assume the total distance as d (as distance does not need to be numerically calculated for this problem), hence, average speed can be calculated as \(v_{avg} = \frac {d}{t_{total}} = \frac {d}{9s} \). This would be the final solution, assuming we have a value for 'd'.

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Key Concepts

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

Understanding Kinematics
Kinematics is a branch of physics that deals with the motion of objects without considering the forces that cause this motion. In simple terms, when we talk about kinematics, we are interested in the 'how' of motion 鈥� how objects move, how fast they go, how far they travel, and how quickly they change their velocity. Key aspects of kinematics include displacement (change in position), speed (how fast an object is moving), velocity (speed in a given direction), and acceleration (how quickly the velocity changes).

One of the most fundamental concepts in kinematics is the average speed. The average speed is calculated by dividing the total distance traveled by the total time taken to travel that distance. Understanding the average speed is crucial as it gives you the 'big picture' of a particle's motion over a period, irrespective of its varying speeds at different intervals.
Analyzing Particle Motion
Particle motion refers to the movement of a particle along a path over time. It's an essential concept in physics and engineering where we model objects as particles to simplify the analysis of their movement. When a particle changes its position, it is said to be in motion. The path can be linear, circular, or any arbitrary curve, and the time it takes to go from one point to another is a significant factor in studying this motion.

In the context of the given exercise, the particle鈥檚 path includes segments from point A to B, B to C, and C to D, with varying times for each segment. To properly analyze this motion, we would study the speed at each part of the journey and how the particle accelerates or decelerates. For an exercise-oriented approach, focusing on the average speed simplifies this complexity and provides a quick way to gauge the particle's overall motion from A to D.
Calculating Time of Travel
Time of travel is a straightforward yet vital concept in kinematics; it's simply how long an object takes to move from one point to another. This duration is critical for calculating other kinematic quantities such as speed and acceleration. In our example, the times taken for the particle to travel between points A to B, B to C, and C to D are all different, highlighting that the particle may be speeding up or slowing down at different parts of the journey.

To get a complete understanding of the particle's motion, we accumulate the total time taken. Here, the sum of the individual times gives us a useful piece of information: the total time of travel. This can be directly applied to find the average speed, as it is the denominator in the average speed formula, \( v_{avg} = \frac {d}{t_{total}} \). Hence, summarizing the time of travel accurately is essential for such calculations.

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Most popular questions from this chapter

The motion of a particle is defined by the equations \(x=\left(2 t+t^{2}\right) \mathrm{m}\) and \(y=\left(t^{2}\right) \mathrm{m},\) where \(t\) is in seconds. Determine the normal and tangential components of the particle's velocity and acceleration when \(t=2 \mathrm{~s}\).

The ball is ejected horizontally from the tube with a speed of \(8 \mathrm{~m} / \mathrm{s}\). Find the equation of the path, \(y=f(x)\) and then find the ball's velocity and the normal and tangential components of acceleration when \(t=0.25 \mathrm{~s}\).

When a rocket reaches an altitude of \(40 \mathrm{~m}\) it begins to travel along the parabolic path \((y-40)^{2}=160 x,\) where the coordinates are measured in meters. If the component of velocity in the vertical direction is constant at \(v_{y}=180 \mathrm{~m} / \mathrm{s}\) determine the magnitudes of the rocket's velocity and acceleration when it reaches an altitude of \(80 \mathrm{~m}\).

The projectile is launched with a velocity \(\mathbf{v}_{0}\). Determine the range \(R,\) the maximum height \(h\) attained, and the time of flight. Express the results in terms of the angle \(\theta\) and \(v_{0}\). The acceleration due to gravity is \(g\).

The partial surface of the cam is that of a logarithmic spiral \(r=\left(40 e^{0.05 \theta}\right) \mathrm{mm},\) where \(\theta\) is in radians. If the cam is rotating at a constant angular rate of \(\dot{\theta}=4 \mathrm{rad} / \mathrm{s}\), determine the magnitudes of the velocity and acceleration of the follower rod at the instant \(\theta=30^{\circ}\).
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