/*! 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 80 Two motorcycles are traveling du... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 80

Two motorcycles are traveling due east with different velocities. However, four seconds later, they have the same velocity. During this four-second interval, motorcycle A has an average acceleration of \(2.0 \mathrm{~m} / \mathrm{s}^{2}\) due east, while motorcycle \(\mathrm{B}\) has an average acceleration of \(4.0 \mathrm{~m} / \mathrm{s}^{2}\) due east. By how much did the speeds differ at the beginning of the four-second interval, and which motorcycle was moving faster?

Short Answer

Expert verified
Motorcycle A was initially faster by 8 m/s.

Step by step solution

01

Identify Initial Conditions and Formulas

Both motorcycles initially have different velocities, but they equalize after 4 seconds. Use the formula for final velocity in uniformly accelerated rectilinear motion: \( v_f = v_i + a \cdot t \), where \( v_f \) is the final velocity, \( v_i \) is the initial velocity, \( a \) is acceleration, and \( t \) is time.
02

Setup Equations for Final Velocities

For motorcycle A, the final velocity equation is: \( v_{f,A} = v_{i,A} + 2.0 \cdot 4 \). For motorcycle B, the final velocity equation is: \( v_{f,B} = v_{i,B} + 4.0 \cdot 4 \). Since \( v_{f,A} = v_{f,B} \) after 4 seconds, set these equations equal to each other.
03

Equate the Final Velocities

Equating the final velocities gives: \( v_{i,A} + 8 = v_{i,B} + 16 \). Simplifying, we find \( v_{i,B} = v_{i,A} - 8 \). This shows that A started 8 m/s faster than B.
04

Subtract to Find the Initial Speed Difference

Since \( v_{i,B} = v_{i,A} - 8 \), the difference in their initial speeds is 8 m/s. Therefore, motorcycle A's initial speed was greater by 8 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.

Uniform Acceleration
Understanding uniform acceleration is a key part of solving motion problems. Uniform acceleration means that the acceleration is constant over a period. In simpler terms, the velocity of an object changes by the same amount every second. This concept is crucial for predicting how the velocity of an object will change over time.
In the exercise, both motorcycles experience uniform acceleration but with different magnitudes:
  • Motorcycle A has a uniform acceleration of 2.0 m/s虏
  • Motorcycle B has a uniform acceleration of 4.0 m/s虏
Knowing this helps us use the right equations to calculate changes in velocity. Uniform acceleration makes motion calculations predictable and straightforward, because the variables involved maintain a steady relationship with time.
Velocity Equations
Velocity equations are essential tools in kinematics for understanding the motion of objects. They allow us to calculate how velocity changes over time, given an object's acceleration and initial velocity. The basic formula used here is:
  • \( v_f = v_i + a \cdot t \)
This equation tells us how the final velocity \( v_f \) depends on the initial velocity \( v_i \), the acceleration \( a \), and the time duration \( t \) of the acceleration.
In our problem, we applied this equation to both motorcycles to express their final velocities after four seconds of acceleration. These expressions helped us determine that both motorcyclists had equal velocities at the end of this period. The power of velocity equations lies in their ability to simplify complex motion scenarios into manageable computations.
Initial Velocity
Initial velocity is the velocity of an object before any changes in speed due to acceleration. It's the starting speed considered when analyzing motion. Calculating initial velocity is crucial when determining how much acceleration is required to reach a certain final velocity, or in understanding the changes in speed over time.
In the context of our exercise, we were given conditions that led us to focus on calculating the difference in initial velocities between the two motorcycles. By equating the final velocities and knowing the acceleration each motorcycle underwent, we were able to deduce that motorcycle A had an initial velocity 8 m/s higher than motorcycle B. This task illustrates how pivotal the concept of initial velocity is in solving motion problems within kinematics.

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

Review Interactive Solution 2.49 at before beginning this problem. A woman on a bridge \(75.0 \mathrm{~m}\) high sees a raft floating at a constant speed on the river below. She drops a stone from rest in an attempt to hit the raft. The stone is released when the raft has 7.00 \(\mathrm{m}\) more to travel before passing under the bridge. The stone hits the water \(4.00 \mathrm{~m}\) in front of the raft. Find the speed of the raft.

Refer to Concept Simulation 2.4 at for help in visualizing this problem graphically. A cart is driven by a large propeller or fan, which can accelerate or decelerate the cart. The cart starts out at the position \(x=0 \mathrm{~m}\), with an initial velocity of \(+5.0 \mathrm{~m} / \mathrm{s}\) and a constant acceleration due to the fan. The direction to the right is positive. The cart reaches a maximum position of \(x=+12.5 \mathrm{~m},\) where it begins to travel in the negative direction. Find the acceleration of the cart.

A sprinter explodes out of the starting block with an acceleration of \(+2.3 \mathrm{~m} / \mathrm{s}^{2},\) which she sustains for \(1.2 \mathrm{~s}\). Then, her acceleration drops to zero for the rest of the race. What is her velocity (a) at \(t=1.2 \mathrm{~s}\) and \((\mathrm{b})\) at the end of the race?

A speedboat starts from rest and accelerates at \(+2.01 \mathrm{~m} / \mathrm{s}^{2}\) for \(7.00 \mathrm{~s}\). At the end of this time, the boat continues for an additional \(6.00 \mathrm{~s}\) with an acceleration of \(+0.518 \mathrm{~m} / \mathrm{s}^{2}\) Following this, the boat accelerates at \(-1.49 \mathrm{~m} / \mathrm{s}^{2}\) for \(8.00 \mathrm{~s}\). (a) What is the velocity of the boat at \(t=21.0 \mathrm{~s} ?\) (b) Find the total displacement of the boat.

In 1998 , NASA launched Deep Space \(I\) (DS-1), a spacecraft that successfully flew by the asteroid named 1992 KD (which orbits the sun millions of miles from the earth). The propulsion system of DS-1 worked by ejecting high-speed argon ions out the rear of the engine. The engine slowly increased the velocity of DS-1 by about \(+9.0 \mathrm{~m} / \mathrm{s}\) per day. (a) How much time (in days) would it take to increase the velocity of DS-1 by \(+2700 \mathrm{~m} / \mathrm{s}\) ? (b) What was the acceleration of \(\mathrm{DS}-1\left(\mathrm{in} \mathrm{m} / \mathrm{s}^{2}\right) ?\)
See all solutions

Recommended explanations on Physics Textbooks

Linear Momentum

Read Explanation

Oscillations

Read Explanation

Translational Dynamics

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Torque and Rotational Motion

Read Explanation

Measurements

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.