/*! 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 32 A truck on a straight road start... [FREE SOLUTION] | 91Ó°ÊÓ

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A truck on a straight road starts from rest, accelerating at 2.00 \(\mathrm{m} / \mathrm{s}^{2}\) until it reaches a speed of 20.0 \(\mathrm{m} / \mathrm{s}\) . Then the truck travels for 20.0 s at constant speed until the brakes are applied, stopping the truck in a uniform manner in an additional 5.00 s. (a) How long is the truck in motion? (b) What is the average velocity of the truck for the motion described?

Short Answer

Expert verified
The truck is in motion for 35 seconds, and its average velocity is 15.71 m/s for the described motion.

Step by step solution

01

Calculate the time to reach 20 m/s

Using the kinematic equation \(v = u + at\), where \(v\) is the final velocity, \(u\) is the initial velocity (0 m/s in this case since the truck is starting from rest), \(a\) is the acceleration, and \(t\) is the time. Solve for \(t\) to find how long it takes to reach 20 m/s: \(20.0 = 0 + 2.00t\), hence \(t = \frac{20.0}{2.00} = 10 s\).
02

Determine the total time the truck is in motion

Sum the time taken to reach 20 m/s, the time traveled at constant speed, and the time to stop: \(10 s + 20 s + 5 s = 35 s\).
03

Calculate the distance traveled during acceleration

Use the kinematic equation \(s = ut + \frac{1}{2}at^2\), where \(s\) is the distance, \(u\) is initial velocity, \(a\) is acceleration, and \(t\) is the time. Since initial velocity \(u = 0\), the equation simplifies to \(s = \frac{1}{2}at^2\). Substituting the given values, \(s = \frac{1}{2} \cdot 2.00 \cdot (10)^2 = 100 m\).
04

Calculate the distance traveled at constant speed

The distance is found by multiplying the constant speed by the time at that speed: \(s = vt = 20.0 \cdot 20.0 = 400 m\).
05

Calculate the distance during deceleration to a stop

Given that the truck stops uniformly in 5 seconds and the initial speed during this time is 20 m/s, the average speed during deceleration is half of 20 m/s (since it goes from 20 m/s to 0 m/s uniformly). The distance \(s\) is then \(s = \frac{(20 + 0)}{2} \cdot 5 = 50 m\).
06

Calculate the total distance traveled

Sum the distances from each phase of the journey: \(100 m + 400 m + 50 m = 550 m\).
07

Calculate the average velocity

The average velocity is total displacement divided by the total time. Since the direction does not change, average velocity is \(v_{avg} = \frac{Total Distance}{Total Time} = \frac{550 m}{35 s} = 15.71 m/s\).

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Key Concepts

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

Uniform Acceleration
Uniform acceleration occurs when an object's acceleration, or its rate of change in velocity, is constant over time. In the example of the truck, acceleration is given as 2.00 m/s², which means the truck's speed increases by 2.00 meters per second every second until it reaches its final velocity of 20.0 m/s.

Understanding this concept is crucial when dealing with kinematic equations, as it allows for the prediction of other motion parameters such as distance traveled and time taken to reach a certain speed. The equation used in the given solution,
\( v = u + at \),
is a direct application of the concept of uniform acceleration where \(v\) is the final velocity, \(u\) is the initial velocity, \(a\) is the constant acceleration, and \(t\) is the time elapsed.
Average Velocity
Average velocity is defined as the total displacement divided by the total time taken to travel that displacement. It is a vector quantity, meaning it has both magnitude and direction. However, in the case of the truck moving in a straight line, the direction remains constant.

In the step-by-step solution, the average velocity is calculated in the final step. Since the truck travels in one direction, the average velocity is simplified as the ratio of the total distance traveled to the total time of travel. This yields the formula:
\( v_{avg} = \frac{Total\ Distance}{Total\ Time} \).
Calculating average velocity is essential as it provides a single value that summarizes the overall motion of the truck along the straight road.
Constant Speed Motion
Constant speed motion implies that an object travels at a uniform speed over a period of time, without speeding up or slowing down. During this phase of motion, the acceleration is zero. For the truck exercise, this is seen during the 20-second interval where it travels at a steady speed of 20.0 m/s.

To find the distance traveled during this phase, a simple calculation of speed multiplied by time is used. The formula \(s = vt\) clearly demonstrates this relationship. Here, \(s\) represents the distance, \(v\) the constant speed, and \(t\) the time interval during which the speed is constant.
Distance Traveled
The distance traveled is the total length of the path covered by a moving object. This concept is crucial for solving many kinematics problems. The truck's total distance traveled is the sum of the distances covered during each phase of its journey: acceleration, constant speed, and deceleration.

During acceleration and deceleration, kinematic equations that account for the changing velocity are used, such as \(s = ut + \frac{1}{2}at^2\) for acceleration and a similar approach for deceleration considering the average speed. For constant speed motion, the simple formula \(s = vt\) is sufficient. By adding the distances from all phases, we obtain the total distance traveled by the truck throughout its entire journey.

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Most popular questions from this chapter

An object moves along the \(x\) axis according to the equation \(x(t)=\left(3.00 t^{2}-2.00 t+3.00\right) \mathrm{m} .\) Determine (a) the aver- age speed between \(t=2.00 \mathrm{s}\) and \(t=3.00 \mathrm{s},(\mathrm{b})\) the instantaneous speed at \(t=2.00 \mathrm{s}\) and at \(t=3.00 \mathrm{s},(\mathrm{c})\) the average acceleration between \(t=2.00 \mathrm{s}\) and \(t=3.00 \mathrm{s},\) and \((\mathrm{d})\) the instantaneous acceleration at \(t=2.00 \mathrm{s}\) and \(t=3.00 \mathrm{s}\)

A test rocket is fired vertically upward from a well. A catapult gives it an initial speed of 80.0 \(\mathrm{m} / \mathrm{s}\) at ground level. Its engines then fire and it accelerates upward at 4.00 \(\mathrm{m} / \mathrm{s}^{2}\) until it reaches an altitude of 1000 \(\mathrm{m}\) . At that point its engines fail and the rocket goes into free fall, with an acceleration of \(-9.80 \mathrm{m} / \mathrm{s}^{2}\) . (a) How long is the rocket in motion above the ground? (b) What is its maximum altitude? (c) What is its velocity just before it collides with the Earth? (You will need to consider the motion while the engine is operating separate from the free-fall motion.)

A woman is reported to have fallen 144 ft from the 17 th floor of a building, landing on a metal ventilator box, which she crushed to a depth of 18.0 in. She suffered only minor injuries. Neglecting air resistance, calculate (a) the speed of the woman just before she collided with the ventilator, (b) her average acceleration while in contact with the box, and (c) the time it took to crush the box.

Automotive engineers refer to the time rate of change of acceleration as the "jerk." If an object moves in one dimension such that its jerk \(J\) is constant, (a) determine expressions for its acceleration \(a_{x}(t),\) velocity \(v_{x}(t),\) and position \(x(t),\) given that its initial acceleration, velocity, and position are \(a_{x i}, v_{x i},\) and \(x_{i},\) respectively. (b) Show that \(a_{x}^{2}=\) \(a_{x i}^{2}+2 J\left(v_{x}-v_{x i}\right) .\)

The position of a particle moving along the \(x\) axis varies in time according to the expression \(x=3 t^{2},\) where \(x\) is in meters and \(t\) is in seconds. Evaluate its position \((a)\) at \(t=3.00 \mathrm{s}\) and \((\mathrm{b})\) at \(3.00 \mathrm{s}+\Delta t .\) (c) Evaluate the limit of \(\Delta x / \Delta t\) as \(\Delta t\) approaches zero, to find the velocity at \(t=3.00 \mathrm{s}\) .

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