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Chapter 6: Problem 82

Some gliders are launched from the ground by means of a winch, which rapidly reels in a towing cable attached to the glider. What average power must the winch supply in order to accelerate a 184 -kg ultralight glider from rest to \(26.0 \mathrm{m} / \mathrm{s}\) over a horizontal distance of \(48.0 \mathrm{m} ?\) Assume that friction and air resistance are negligible, and that the tension in the winch cable is constant.

Short Answer

Expert verified
The winch must supply about 16834 Watts of power.

Step by step solution

01

Understand the Given

We are tasked with finding the average power needed by a winch to accelerate a glider from rest to a certain velocity over a specified distance. The mass of the glider is given as 184 kg, the final velocity is 26.0 m/s, and the horizontal distance covered is 48.0 meters. We assume no friction or air resistance.
02

Calculate the Work Done

The work done by the winch equals the change in kinetic energy of the glider. Use the kinetic energy formula: \( KE = \frac{1}{2} mv^2 \). Initially, the kinetic energy is 0 since the glider is at rest. The work done by the winch is the final kinetic energy:\[W = \frac{1}{2} \times 184 \times (26.0)^2 = 62168 \text{ J}\]
03

Calculate the Time Taken

To find power, we need to know the time. Since we only know the distance and speeds, use the formula for work done over distance, rearranged from the definition of average acceleration \( a \) and velocity \( v \):\[ v^2 = 2as \]\[ a = \frac{v^2}{2s} = \frac{(26.0)^2}{2 \times 48.0} = 7.042 \text{ m/s}^2\]Use acceleration to find time \( t \) using \( v = at \):\[ t = \frac{v}{a} = \frac{26.0}{7.042} \approx 3.693 \text{ s} \]
04

Calculate the Average Power

Calculate the average power using the work done and the time:\[ P = \frac{W}{t} = \frac{62168}{3.693} \approx 16834 \text{ W} \]Thus, the average power supplied by the winch is approximately 16834 Watts.

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Key Concepts

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

Kinetic Energy
Kinetic energy is the energy an object possesses due to its motion. Think about it this way: when anything moves, it carries an energy package called kinetic energy. For a moving object, this energy is determined by both its mass and its speed. The formula to calculate it is given by \[ KE = \frac{1}{2} mv^2 \] where \( m \) is the mass of the object and \( v \) is its velocity. In our exercise, the ultralight glider is moving from rest, so initially, it has zero kinetic energy. As the winch accelerates it to a speed of 26.0 m/s, the kinetic energy changes. By plugging in the values, we calculated that the final kinetic energy of the glider is 62,168 Joules. This value represents the work done by the winch to propel the glider forward.
Work-Energy Principle
The work-energy principle provides a deep insight into studying physics problems related to energy. This principle states that the work done by all forces acting on an object equals the change in its kinetic energy. To put it simply, any work done on an object translates into its energy change.
  • "Work done" is essentially a force acting over a distance.
  • "Change in kinetic energy" is the difference between the object's final and initial kinetic energy.
In our exercise, the winch's purpose was to work the glider from a stationary state to a high speed. By understanding this, we realize that the 62,168 Joules of work done by the winch corresponds to the glider's kinetic energy change from rest.
Constant Acceleration
When an object moves with constant acceleration, it means its velocity changes steadily over time. This concept is crucial in solving many physics problems, particularly those involving motion calculations. Using the formula \[ v^2 = 2as \] we can determine how this constant acceleration impacts other variables, such as velocity and distance. Here, starting from rest, the glider achieved a final speed of 26.0 m/s over a 48.0-meter distance. By plugging these numbers into the formula and solving for acceleration \( a \), we found the value to be approximately 7.042 m/s虏.

Once the acceleration is known, we can find out how long this process took. Using the equation \[ v = at \] in which \( t \) is the time, we further determined that it took around 3.693 seconds for the acceleration process.
Physics Problems
Physics problems often require us to synthesize different concepts to find a solution. They involve understanding the problem parameters, identifying relevant equations, and integrating knowledge from various fields. Here, to calculate the average power output of a winch:
  • We started by computing the change in kinetic energy to find the total work done.
  • Then calculated the constant acceleration and time required using known equations.
  • Finally, we determined the average power by dividing work done by time.
Bringing all these computations together provides a complete solution. Understanding the links between kinetic energy, work, acceleration, and time becomes crucial for forming accurate conclusions in such physics problems. By breaking it down into smaller parts and focusing on each component, we can effectively solve and understand complex physics challenges.

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Most popular questions from this chapter

A helicopter, starting from rest, accelerates straight up from the roof of a hospital. The lifting force does work in raising the helicopter. An \(810-\mathrm{kg}\) helicopter rises from rest to a speed of \(7.0 \mathrm{m} / \mathrm{s}\) in a time of \(3.5 \mathrm{s}\) During this time it climbs to a height of \(8.2 \mathrm{m} .\) What is the average power generated by the lifting force?

A small lead ball, attached to a 0.75-m rope, is being whirled in a circle that lies in the vertical plane. The ball is whirled at a constant rate of three revolutions per second and is released on the upward part of the circular motion when it is \(1.5 \mathrm{m}\) above the ground. The ball travels straight upward. In the absence of air resistance, to what maximum height above the ground does the ball rise?

In 2.0 minutes, a ski lift raises four skiers at constant speed to a height of \(140 \mathrm{m}\). The average mass of each skier is \(65 \mathrm{kg}\). What is the average power provided by the tension in the cable pulling the lift?

The drawing shows two frictionless inclines that begin at ground level \((h=0 \mathrm{m})\) and slope upward at the same angle \(\theta .\) One track is longer than the other, however. Identical blocks are projected up each track with the same initial speed \(v_{0}\). On the longer track the block slides upward until it reaches a maximum height \(H\) above the ground. On the shorter track the block slides upward, flies off the end of the track at a height \(H_{1}\) above the ground, and then follows the familiar parabolic trajectory of projectile motion. At the highest point of this trajectory, the block is a height \(H_{2}\) above the end of the track. The initial total mechanical energy of each block is the same and is all kinetic energy. The initial speed of each block is \(v_{0}=7.00 \mathrm{m} / \mathrm{s},\) and each incline slopes upward at an angle of \(\theta=50.0^{\circ} .\) The block on the shorter track leaves the track at a height of \(H_{1}=1.25 \mathrm{m}\) above the ground. Find (a) the height \(H\) for the block on the longer track and (b) the total height \(H_{1}+H_{2}\) for the block on the shorter track.

A Sledding Contest. You are in a sledding contest where you start at a height of \(40.0 \mathrm{m}\) above the bottom of a valley and slide down a hill that makes an angle of \(25.0^{\circ}\) with respect to the horizontal. When you reach the valley, you immediately climb a second hill that makes an angle of \(15.0^{\circ}\) with respect to the horizontal. The winner of the contest will be the contestant who travels the greatest distance up the second hill. You must now choose between using your flat-bottomed plastic sled, or your "Blade Runner," which glides on two steel rails. The hill you will ride down is covered with loose snow. However, the hill you will climb on the other side is a popular sledding hill, and is packed hard and is slick. The two sleds perform very differently on the two surfaces, the plastic one performing better on loose snow, and the Blade Runner doing better on hard-packed snow or ice. The performances of each sled can be quantified in terms of their respective coefficients of kinetic friction on the two surfaces. For the plastic sled: \(\mu=0.17\) on loose snow, and \(\mu=0.15\) on packed snow or ice. For the Blade Runner, \(\mu=0.19\) on loose snow, and \(\mu=0.07\) on packed snow or ice. Assuming the two hills are shaped like inclined planes, and neglecting air resistance, (a) how far does each sled make it up the second hill before stopping? (b) Assuming the total mass of the sled plus rider is \(55.0 \mathrm{kg}\) in both cases, how much work is done by nonconservative forces (over the total trip) in each case?
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