/*! 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 13 For a short period of time, the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 15: Problem 13

For a short period of time, the frictional driving force acting on the wheels of the \(2.5-\mathrm{Mg}\) van is \(F_{D}=\left(600 t^{2}\right) \mathrm{N}\), where \(t\) is in seconds. If the van has a speed of \(20 \mathrm{~km} / \mathrm{h}\) when \(t=0,\) determine its speed when \(t=5 \mathrm{~s}\)

Short Answer

Expert verified
The speed of the van after \(t = 5 s\) is \(35.56 m/s.\)

Step by step solution

01

Calculate the driving force after 5 seconds

From the formula \(F_{D} = 600 t^{2}\) N, substitute \(t = 5\) s into the equation to find the driving force at that time. Thus, \(F_{D} = 600 \cdot (5)^{2} = 15000\) N.
02

Calculate the acceleration

Using Newton's second law, we can find the acceleration of the van by rearranging \(F = ma\) to get \(a = F/m\). Substituting in the values, we get, \(a = F_{D}/m = 15000 / 2500 = 6 \,m/s^{2}\).
03

Calculate the change in velocity

The change in velocity can be calculated using the formula \(\Delta v = a \cdot \Delta t\). Substituting in the acceleration from step 2 and the time of \(5 s\), we get \(\Delta v = 6 \cdot 5 = 30 \,m/s\).
04

Convert the initial speed to m/s

The initial speed of the van is given in km/h. To make it compatible with other quantities in the problem, convert it to m/s using the conversion factor 1 km/h = \(5/18 \,m/s\). Thus, the initial speed, \(v_{i} = 20 \cdot (5/18) = 5.56 \,m/s\).
05

Calculate the final speed

The final speed of the van can be found by adding the initial speed to the change in speed: \(v_{f} = v_{i} + \Delta v = 5.56 \,m/s + 30 \,m/s = 35.56 \,m/s\).

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 91影视!

Key Concepts

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

Newton's second law
Understanding Newton's second law is crucial when it comes to dynamics problem solving. This law states that the acceleration of an object is directly proportional to the net force acting upon it and inversely proportional to its mass, which can be expressed with the formula:
\[ F = ma \]
where \( F \) is the net force applied to the object, \( m \) is the mass of the object, and \( a \) is the acceleration. In our exercise, the frictional driving force is the net force propelling the van forward. By rearranging the formula to solve for acceleration, \( a = \frac{F}{m} \), we can determine how the van's velocity will change over time as a result of the force applied.
Acceleration calculation
Acceleration is the rate of change of velocity of an object. It is a vector quantity, which means it has both magnitude and direction. To calculate the acceleration, you need to know the net force acting on the object and its mass. The formula provided by Newton's second law, \( a = \frac{F}{m} \), allows us to find the acceleration.

Using the step by step solution from our exercise, once the driving force \( F_D \) is known, by substituting it along with the mass of the van into the formula, the acceleration can be calculated. In our example, the force is a function of time, which adds a layer of complexity. For different moments in time, the driving force changes, and consequently, so does the acceleration.
Velocity conversion
Velocity conversion between units is a basic yet vital skill in physics problems, especially in dynamics where consistency of units is key. The standard unit for velocity in the International System of Units (SI) is meters per second (m/s). However, it is common to encounter velocities given in kilometers per hour (km/h). To convert from km/h to m/s, you can multiply the speed by the conversion factor \( \frac{5}{18} \).

In our exercise, the initial velocity is given as \(20 \) km/h which we convert to \(5.56 \) m/s before it can be used for further calculations. This step is essential to ensure that when we calculate the final speed, by adding the change in velocity due to the driving force, all variables are in the correct and consistent unit of measure.

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

At the instant \(r=1.5 \mathrm{~m}\), the \(5-\mathrm{kg}\) disk is given a speed of \(v=5 \mathrm{~m} / \mathrm{s}\), perpendicular to the elastic cord. Determine the speed of the disk and the rate of shortening of the elastic cord at the instant \(r=1.2 \mathrm{~m}\). The disk slides on the smooth horizontal plane. Neglect its size. The cord has an unstretched length of \(0.5 \mathrm{~m}\).

The cue ball \(A\) is given an initial velocity \(\left(v_{A}\right)_{1}=5 \mathrm{~m} / \mathrm{s}\). If it makes a direct collision with ball \(B(e=0.8)\), determine the velocity of \(B\) and the angle \(\theta\) just after it rebounds from the cushion at \(C\left(e^{\prime}=0.6\right)\). Each ball has a mass of \(0.4 \mathrm{~kg}\). Neglect their size.

The 50 -kg crate is pulled by the constant force \(\mathbf{P}\). If the crate starts from rest and achieves a speed of \(10 \mathrm{~m} / \mathrm{s}\) in \(5 \mathrm{~s}\), determine the magnitude of \(\mathbf{P}\). The coefficient of kinetic friction between the crate and the ground is \(\mu_{k}=0.2\).

The \(2.5-\mathrm{Mg}\) van is traveling with a speed of \(100 \mathrm{~km} / \mathrm{h}\) when the brakes are applied and all four wheels lock. If the speed decreases to \(40 \mathrm{~km} / \mathrm{h}\) in \(5 \mathrm{~s}\), determine the coefficient of kinetic friction between the tires and the road.

An earth satellite of mass \(700 \mathrm{~kg}\) is launched into a free-flight trajectory about the earth with an initial speed of \(v_{A}=10 \mathrm{~km} / \mathrm{s}\) when the distance from the center of the earth is \(r_{A}=15 \mathrm{Mm}\). If the launch angle at this position is \(\phi_{A}=70^{\circ},\) determine the speed \(v_{B}\) of the satellite and its closest distance \(r_{B}\) from the center of the earth. The earth has a mass \(M_{e}=5.976\left(10^{24}\right) \mathrm{kg} .\) Hint: Under these conditions, the satellite is subjected only to the earth's gravitational force, \(F=G M_{e} m_{s} / r^{2},\) Eq. \(13-1\). For part of the solution, use the conservation of energy.
See all solutions

Recommended explanations on Physics Textbooks

Mechanics and Materials

Read Explanation

Fluids

Read Explanation

Space Physics

Read Explanation

Oscillations

Read Explanation

Electricity

Read Explanation

Medical Physics

Read Explanation
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.