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 163-kg ultralight glider from rest to 27.9 m/s over a horizontal distance of 46.8 m? Assume that friction and air resistance are negligible, and that the tension in the winch cable is constant.

power=1/2 m 27.9^2 /time

where time= distance*2/27.9

Why did the glider become a stand-up comedian? Because it wanted to fly high in the sky and tell jokes that would make everyone soar with laughter! Now, let's calculate the average power required by the winch.

We can start by calculating the work done to accelerate the glider. The work done is equal to the change in kinetic energy. The formula for kinetic energy is ½mv², where m is the mass of the glider and v is the final velocity.

The change in kinetic energy is given by:

ΔKE = ½mv² - ½m(0)² = ½mv²

So, the work done to accelerate the glider is equal to ½mv². We can now calculate the average power using the formula for power:

Power = work done / time

We know the distance traveled (46.8 m) and the final velocity (27.9 m/s). However, we need to find the time taken to cover this distance. To do that, we can use the formula:

distance = average velocity × time

Since the initial velocity is 0 m/s, the average velocity is half of the final velocity.

Using this equation, we can calculate the time:

46.8 m = (½ × 27.9 m/s) × time

Simplifying this equation, we find:

time = 46.8 m / (½ × 27.9 m/s)

Now we can calculate the average power:

Power = (½ × 163 kg × (27.9 m/s)²) / (46.8 m / (½ × 27.9 m/s))

Let's do the math:

Power ≈ 124,652.78 watts

So, the winch needs to supply approximately 124,652.78 watts of power to accelerate the glider. That's a lot of power, but it's worth it for a glider seeking to take flight!

To find the average power supplied by the winch, we can use the work-energy theorem. The work done on the glider is equal to its change in kinetic energy:

Work = change in kinetic energy

Since the glider starts from rest, the change in kinetic energy is equal to its final kinetic energy:

Work = (1/2)mv^2

where m is the mass of the glider (163 kg) and v is the final velocity (27.9 m/s).

The work done is also equal to the force applied by the winch multiplied by the distance over which it acts:

Work = Force x Distance

The force applied by the winch is the tension in the cable, and the distance is the horizontal distance traveled by the glider (46.8 m).

Therefore,

(1/2)mv^2 = Force x Distance

Solving for the force:

Force = (1/2)mv^2 / Distance

Substituting the given values:

Force = (1/2)(163 kg)(27.9 m/s)^2 / 46.8 m

Calculating the force:

Force = (1/2)(163 kg)(778.41 m^2/s^2) / 46.8 m

Force = 6387.88 N

Finally, the average power supplied by the winch is the force multiplied by the glider's velocity:

Average Power = Force x Velocity

Average Power = 6387.88 N x 27.9 m/s

Calculating the average power:

Average Power ≈ 178,043.65 Watts

To determine the average power that the winch must supply to accelerate the glider, we can use the work-energy principle. The work done on an object is equal to the change in its kinetic energy.

The work done can be calculated using the formula:

Work = Force * Distance * cosine(theta),

where the force is the tension in the cable and theta is the angle between the direction of the force and the direction of motion.

Since the glider is accelerated horizontally, the angle between the force and the direction of motion is 0 degrees. Therefore, cosine(theta) = 1.

The force is equal to the mass of the glider times its acceleration:

Force = mass * acceleration.

The distance is given as 46.8 m, and the glider's mass is 163 kg.

First, we need to find the acceleration of the glider. Using the kinematic equation:

v^2 = u^2 + 2as,

where v is the final velocity, u is the initial velocity (0 m/s in this case since the glider is at rest), and s is the distance, we can rearrange the equation to find the acceleration:

a = (v^2 - u^2) / (2s),

a = (27.9^2 - 0^2) / (2 * 46.8).

a ≈ 19.5 m/s^2.

Now, we can calculate the force:

Force = mass * acceleration,

Force = 163 kg * 19.5 m/s^2.

Force ≈ 3178.5 N (Newtons).

Since cosine(theta) is 1, we can calculate the work done:

Work = Force * Distance,

Work = 3178.5 N * 46.8 m.

Work ≈ 148,622.8 Joules.

Finally, we can calculate the average power by dividing the work done by the time taken:

Average Power = Work / Time.

Since the time is not given, we can use the average velocity to find the time:

Time = Distance / Average Velocity,

Time = 46.8 m / (27.9 m/s / 2).

Time ≈ 3.3 seconds.

Average Power = 148,622.8 Joules / 3.3 seconds.

Average Power ≈ 45,065.9 Watts.

Therefore, the winch must supply an average power of approximately 45,065.9 Watts to accelerate the glider.