91Ó°ÊÓ

Under the influence of its drive force, a snowmobile is moving at a constant velocity along a horizontal patch of snow. When the drive force is shut off, the snowmobile coasts to a halt. (a) During the coasting phase, what is the net external force acting on the snowmobile? (b) How does the work \(W\) done by the net external force depend on the magnitude and direction of the net force and on the displacement of the snowmobile? (c) What is the net force acting on the snowmobile during the constantvelocity phase? Explain. (d) The work-energy theorem is given by \(W=\frac{1}{2} m v_{\mathrm{f}}^{2}-\frac{1}{2} m v_{0}^{2}\) (Equation 6.3 ), where \(W\) is the work done by the net external force acting on the snowmobile, \(m\) is its mass, and \(v_{\mathrm{f}}\) and \(v_{0}\) are, respectively, its final and initial speeds. Suppose the net external force, \(m,\) and \(v_{\mathrm{f}}\) and \(v_{0}\) are known. By using only the work-energy theorem, can one determine the time \(t\) it takes for the snowmobile to coast to a halt? The snowmobile and its rider have a mass of \(136 \mathrm{~kg}\). Under the influence of a drive force of \(205 \mathrm{~N}\), it is moving at a constant velocity, whose magnitude is \(5.50 \mathrm{~m} / \mathrm{s}\). Its drive force is shut off. Find (a) the distance in which the snowmobile coasts to a halt and (b) the time required to do so.

Step by step solution

01

Identify the Forces During Coasting

When the snowmobile coasts to a halt, the driving force is removed. The only force acting on it is the frictional force, which is equal and opposite to its velocity direction. Thus, the net external force is this frictional force, causing the snowmobile to decelerate.

02

Understand the Work Done During Coasting

The work done by the net external force over a displacement is calculated using the equation: \[ W = F \times d \times \cos(\theta) \]where \( F \) is the magnitude of the force, \( d \) is the displacement, and \( \theta \) is the angle between the force and displacement vectors. Since the force is opposed to the direction of motion, \( \theta = 180^\circ \) and \( \cos(\theta) = -1 \).

03

Determine the Net Force During Constant Velocity

During constant velocity, the net force is zero since acceleration is zero, according to Newton's first law. The driving force balances the frictional force.

04

Analyze Work-Energy Theorem for Coasting

Using the work-energy theorem, \( W = \frac{1}{2} m v_{f}^{2} - \frac{1}{2} m v_{0}^{2} \), with \( v_{f} = 0 \) and knowing \( v_{0} = 5.50 \: \mathrm{m/s}\), find the initial kinetic energy and use it to find the work done, which is equal to the negative of the initial kinetic energy.

05

Calculate the Displacement

The work done by the frictional force \( W = F_{friction} \cdot d \) is equal to the initial kinetic energy. Solving for \( d \), where \( F_{friction} = 205 \: \mathrm{N} \) (as per equilibrium during constant velocity):\[ \frac{1}{2} m v_{0}^{2} = 205 \cdot d \]\[ d = \frac{1/2 \cdot 136 \cdot (5.50)^2}{205} \approx 9.17 \: \mathrm{m} \]

06

Calculate the Time to Coast to a Halt

With constant deceleration, use \( v = u + at \) and \( v^2 = u^2 + 2as \) to find acceleration:\[ 0 = (5.50)^2 + 2 \times (-a) \times 9.17 \]\[ a = \frac{(5.50)^2}{2 \times 9.17} \approx 1.65 \: \mathrm{m/s^2} \]Subsequently, use \( v = u + at \):\[ 0 = 5.50 - 1.65 \times t \]\[ t = \frac{5.50}{1.65} \approx 3.33 \: \mathrm{s} \]

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 91Ó°ÊÓ!

Key Concepts

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

Newton's Laws of Motion

Newton's Laws of Motion are the foundation of classical mechanics and help us understand how objects behave under the influence of various forces. These laws are essential for analyzing the movement of the snowmobile in our problem.

• **First Law (Law of Inertia)**: An object at rest stays at rest and an object in motion continues in motion at constant velocity unless acted upon by a net external force. This is observed when the snowmobile moves at a constant velocity because the driving force equals the frictional force.
• **Second Law (Law of Acceleration)**: The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass, expressed as \( F = ma \). This law is relevant when the snowmobile starts decelerating due to the frictional force after the drive force is removed.
• **Third Law (Action-Reaction)**: For every action, there is an equal and opposite reaction. This concept highlights how the frictional force acts against the snowmobile's motion, eventually bringing it to a halt.

Work-Energy Theorem

The Work-Energy Theorem is a powerful concept that connects the work done by forces acting on an object to its kinetic energy change. According to this theorem:\[ W = \frac{1}{2} m v_{f}^{2} - \frac{1}{2} m v_{0}^{2} \]When we apply it to the snowmobile problem, we can observe:

• **Work Done**: The work \( W \) done by the frictional force brings the snowmobile to rest, which is a negative change in kinetic energy.
• With an initial speed \( v_{0} = 5.50 \, \mathrm{m/s} \), the final speed \( v_{f} \) being zero signifies that all the initial kinetic energy has been converted to work done against friction.
• **Energy Conversion**: Initially, all energy is kinetic because of movement, but as the snowmobile slows, energy transfers to perform work against friction.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, defined by the equation:\[ KE = \frac{1}{2} m v^{2} \]For the snowmobile example:

• Initially, the snowmobile moving with a velocity of \( 5.50 \, \mathrm{m/s} \) has a certain amount of kinetic energy associated with its mass and speed.
• The **mass**, here \( 136 \, \mathrm{kg} \), and the initial velocity determine the initial kinetic energy to be converted into work as it slows down.
• As the snowmobile coasts, its kinetic energy decreases due to the frictional force until it becomes zero when the velocity is reduced to zero.

Understandably, the decrease in kinetic energy precisely quantifies how much work is done by frictional forces.

Frictional Force

Frictional force is a resistive force that opposes motion, playing a critical role in stopping the snowmobile. There are important attributes to note:

• **Nature of Friction**: Frictional forces act opposite the direction of motion and are primarily caused by surface interactions.
• In the case of the snowmobile, when coasting, it is the only significant force acting on the snowmobile, gradually reducing its speed to zero. Without any additional forces, this resistive force ensures the snowmobile eventually stops.
• **Constant Velocity Implication**: Before the drive force is turned off, the friction balances the drive force, leading to a net force of zero as per Newton's first law, thus maintaining constant velocity.

By understanding friction, we grasp how it crucially determines the snowmobile's deceleration rate and the total distance covered before stopping.