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Chapter 2: Problem 86

A motorcyclist who is moving along an \(x\) axis directed toward the east has an acceleration given by \(a=(6.1-1.2 t) \mathrm{m} / \mathrm{s}^{2}\) for \(0 \leq t \leq 6.0 \mathrm{~s}\). At \(t=0\), the velocity and position of the cyclist are \(2.7 \mathrm{~m} / \mathrm{s}\) and \(7.3 \mathrm{~m} .\) (a) What is the maximum speed achieved by the cyclist? (b) What total distance does the cyclist travel between \(t=0\) and \(6.0 \mathrm{~s} ?\)

Short Answer

Expert verified
(a) Maximum speed is approximately 18.23 m/s; (b) Total distance is 74.6 m.

Step by step solution

01

Determine the Time of Maximum Speed

The maximum speed occurs when acceleration is zero because before this point the velocity increases, and after this point, it decreases. Set the acceleration equation to zero: \[ 6.1 - 1.2t = 0 \]Solving for \(t\) gives: \[ t = \frac{6.1}{1.2} \approx 5.08 \, \text{s} \]
02

Calculate Maximum Speed

Integrate the acceleration function to find the velocity function. The velocity at any time \(t\) is given by the integral of acceleration:\[ v(t) = \int (6.1 - 1.2t) \, dt = 6.1t - 0.6t^2 + C \]Use the initial condition \(v(0) = 2.7\, \text{m/s}\) to find \(C\):\[ 2.7 = 6.1(0) - 0.6(0)^2 + C \Rightarrow C = 2.7 \]Thus, the velocity function becomes:\[ v(t) = 6.1t - 0.6t^2 + 2.7 \]Substitute \(t = 5.08\, \text{s}\) to find \(v(5.08)\):\[ v(5.08) = 6.1(5.08) - 0.6(5.08)^2 + 2.7 \approx 18.23\, \text{m/s} \]
03

Calculate Total Distance Traveled

Integrate the velocity function to find the position function. The position at any time \(t\) is the integral of velocity:\[ x(t) = \int (6.1t - 0.6t^2 + 2.7) \, dt = 3.05t^2 - 0.2t^3 + 2.7t + D \]Use the initial condition \(x(0) = 7.3\, \text{m}\) to find \(D\):\[ 7.3 = 3.05(0)^2 - 0.2(0)^3 + 2.7(0) + D \Rightarrow D = 7.3 \]Thus, the position function becomes:\[ x(t) = 3.05t^2 - 0.2t^3 + 2.7t + 7.3 \]Calculate \(x(6.0)\) to determine the position at \(t = 6.0\, \text{s}\):\[ x(6.0) = 3.05(6.0)^2 - 0.2(6.0)^3 + 2.7(6.0) + 7.3 = 81.9 \, \text{m} \]The total distance traveled is:\[ x(6.0) - x(0) = 81.9 - 7.3 = 74.6\, \text{m} \]

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Key Concepts

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

Acceleration
Acceleration describes how quickly an object's velocity changes over time. In this problem, the motorcyclist's acceleration along the x-axis is given by the function \( a(t) = 6.1 - 1.2t \, \text{m/s}^2 \). This indicates that the motorcyclist starts with a high acceleration, which then decreases linearly as time goes by. This is a clear example of motion with non-constant acceleration. Contrary to constant acceleration, where the acceleration value remains unchanged, non-constant acceleration involves a varied acceleration rate, leading to complex motion. Understanding these dynamics can help in predicting velocity changes and overall movement along a path.
Velocity
Velocity is a vector quantity describing both the speed and direction of an object's movement. In this exercise, the initial velocity of the motorcyclist is \(2.7 \, \text{m/s} \). To find the velocity at any given time, we integrate the acceleration function. The function yields the velocity equation:
  • \( v(t) = 6.1t - 0.6t^2 + 2.7 \)
To figure out the maximum speed, we set the acceleration to zero, which occurs at \( t \approx 5.08 \, \text{s} \). Evaluating the velocity function at this time gives us the maximum speed of approximately \( 18.23 \, \text{m/s} \). This maximum speed marks the peak in the motorcyclist's motion before deceleration begins because the acceleration function becomes negative beyond this point, indicating a reduction in speed.
Integration in Physics
Integration is a powerful tool used in physics to determine quantities like velocity and position from acceleration. In problems such as these, where acceleration is given, you use integration to calculate velocity and displacement:
  • From acceleration to velocity by integrating the acceleration function.
  • From velocity to position using further integration.
Each integration step involves adding an integration constant, determined by initial conditions given in the problem. Here, integrating the acceleration function \( a(t) \) helps derive the velocity formula, and further integrating the velocity function derives the position formula. These integrated functions provide insights into how the system evolves over time, charting the motorcyclist鈥檚 journey along the x-axis.
Motion Along a Straight Line
Motion along a straight line simplifies the study of kinematics, focusing on movement in one dimension. This exercise places the motorcyclist on an eastward path along the x-axis, so all motion is linear. This means you only need to consider changes in one direction, distinctly simplifying calculations by avoiding vector components.
The motorcyclist鈥檚 motion is influenced by changing acceleration, leading to a straightforward yet dynamic change in both velocity and position over time. Calculating total distance involves finding the position at a future time \( t = 6.0 \, \text{s} \) using the derived position function:
  • \( x(t) = 3.05t^2 - 0.2t^3 + 2.7t + 7.3 \)
By evaluating this function, the total traveled distance is revealed to be approximately \( 74.6 \, \text{m} \), illustrating both the simplicity and depth of linear motion study.

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Most popular questions from this chapter

A lead ball is dropped in a lake from a diving board \(5.20 \mathrm{~m}\) above the water. It hits the water with a certain velocity and then sinks to the bottom with this same constant velocity. It reaches the bottom \(4.80 \mathrm{~s}\) after it is dropped. (a) How deep is the lake? What are the (b) magnitude and (c) direction (up or down) of the average velocity of the ball for the entire fall? Suppose that all the water is drained from the lake. The ball is now thrown from the diving board so that it again reaches the bottom in \(4.80 \mathrm{~s}\). What are the (d) magnitude and (e) direction of the initial velocity of the ball?

Shows part of a street where traffic flow is to be controlled to allow a platoon of cars to move smoothly along the street. Suppose that the platoon leaders have just reached intersection 2, where the green appeared when they were distance \(d\) from the intersection. They continue to travel at a certain speed \(v_{p}\) (the speed limit) to reach intersection 3, where the green appears when they are distance \(d\) from it. The intersections are separated by distances \(D_{23}\) and \(D_{12}\). (a) What should be the time delay of the onset of green at intersection 3 relative to that at intersection 2 to keep the platoon moving smoothly? Suppose, instead, that the platoon had been stopped by a red light at intersection \(1 .\) When the green comes on there, the leaders require a certain time \(t_{r}\) to respond to the change and an additional time to accelerate at some rate \(a\) to the cruising speed \(v_{p} .\) (b) If the green at intersection 2 is to appear when the leaders are distance \(d\) from that intersection, how long after the light at intersection 1 turns green should the light at intersection 2 turn green?

A hot-air balloon is ascending at the rate of \(12 \mathrm{~m} / \mathrm{s}\) and is \(80 \mathrm{~m}\) above the ground when a package is dropped over the side. (a) How long does the package take to reach the ground? (b) With what speed does it hit the ground?

A ball is thrown vertically downward from the top of a \(36.6\) \(\mathrm{m}\) -tall building. The ball passes the top of a window that is \(12.2 \mathrm{~m}\) above the ground \(2.00 \mathrm{~s}\) after being thrown. What is the speed of the ball as it passes the top of the window?

To set a speed record in a measured (straight-line) distance \(d\), a race car must be driven first in one direction (in time \(t_{1}\) ) and then in the opposite direction (in time \(t_{2}\) ). (a) To eliminate the effects of the wind and obtain the car's speed \(v_{c}\) in a windless situation, should we find the average of \(d / t_{1}\) and \(d / t_{2}\) (method 1 ) or should we divide \(d\) by the average of \(t_{1}\) and \(t_{2} ?(\mathrm{~b})\) What is the fractional difference in the two methods when a steady wind blows along the car's route and the ratio of the wind speed \(v_{w}\) to the car's speed \(v_{c}\) is \(0.0240 ?\)
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