A large air-filled 0.1-kg plastic ball is thrown up into the air with an initial speed of 16 m/s. At a height of 4.7 m, the ball's speed is 5 m/s. What fraction of its original energy has been lost to air friction? (Note that the initial height is taken to be zero.)

Initial energy = (M/2)*Vo^2 = 12.80 J

Total energy at 4.7 m and V = 5 m/s:
= (M/2)*5^2 + M*g*4.7 = 1.25 + 4.61 J

Fraction of ball's energy LOST = (12.80-5.86)/12.80
= 0.542

Well, I must say, air friction can be quite the party pooper! Let's calculate how much energy this plastic ball lost while flying through the air, shall we?

The initial kinetic energy of the ball can be calculated using the formula KE = (1/2)mv^2, where m is the mass of the ball (0.1 kg) and v is its initial velocity (16 m/s). So, after crunching the numbers, we find that the initial kinetic energy is 12.8 Joules.

Now, let's find the kinetic energy of the ball when it's at a height of 4.7 m and traveling at a speed of 5 m/s. Again, using the same formula, we get KE = (1/2)mv^2, but this time m is still 0.1 kg and v is 5 m/s. After a little calculation, we find the ball's kinetic energy is 1.25 Joules.

To find the energy lost to air friction, we subtract the final kinetic energy from the initial kinetic energy: 12.8 J - 1.25 J = 11.55 J.

Now, we can determine the fraction of energy lost by dividing the energy lost (11.55 J) by the initial kinetic energy (12.8 J): 11.55 J / 12.8 J ≈ 0.902.

So, it seems like almost 90.2% of the ball's original energy was lost to air friction! Boy, that's a lot of energy getting blown away!

To determine the fraction of the ball's original energy that has been lost to air friction, we'll need to calculate its initial and final kinetic energies.

1. Calculating the initial kinetic energy:
The formula for kinetic energy is given by: KE = (1/2) * m * v^2
Where:
- KE is the kinetic energy in joules (J),
- m is the mass of the ball in kilograms (kg), and
- v is the velocity of the ball in meters per second (m/s).

Plugging in the values:
KE_initial = (1/2) * 0.1 kg * (16 m/s)^2
= 0.1 kg * 8 m^2/s^2
= 0.8 J

2. Calculating the final kinetic energy:
Using the given height, we can find the final velocity of the ball using the conservation of energy principle. The gravitational potential energy at height h is equal to the kinetic energy at the final velocity.

The change in potential energy is given by: ΔPE = m * g * h
Where:
- ΔPE is the change in potential energy in joules (J),
- m is the mass of the ball in kilograms (kg),
- g is the acceleration due to gravity, approximately 9.8 m/s^2, and
- h is the height in meters (m).

Setting the change in potential energy equal to the final kinetic energy:
ΔPE = KE_final

Plugging in the values:
0.1 kg * 9.8 m/s^2 * 4.7 m = KE_final

3. Rearranging the equation to solve for KE_final:
KE_final = 0.1 kg * 9.8 m/s^2 * 4.7 m
= 4.586 J

4. Calculating the fraction of energy lost:
The fraction of energy lost can be found by subtracting the final kinetic energy from the initial kinetic energy and dividing by the initial kinetic energy.
Energy_lost_fraction = (KE_initial - KE_final) / KE_initial
= (0.8 J - 4.586 J) / 0.8 J
≈ -4.786 J / 0.8 J
≈ -5.9825

The fraction of the ball's original energy lost to air friction is approximately -5.9825. Note that a negative value indicates a loss of energy.

To determine the fraction of energy lost to air friction, we first need to find the initial energy and the final energy of the ball.

The initial energy is given by the sum of the kinetic energy (KE) and the potential energy (PE) at the initial height:

Initial Energy = KE(initial) + PE(initial)

The final energy is given by the sum of the kinetic energy and the potential energy at the final height:

Final Energy = KE(final) + PE(final)

First, let's calculate the initial energy:

The initial kinetic energy is given by the formula:

KE(initial) = 0.5 * mass * (velocity)^2

Substituting the given values, we get:

KE(initial) = 0.5 * 0.1 kg * (16 m/s)^2 = 12.8 J

At the initial height, the potential energy is zero since the reference height is taken as zero. Therefore, PE(initial) = 0.

Therefore, the initial energy is:

Initial Energy = KE(initial) + PE(initial) = 12.8 J + 0 J = 12.8 J

Now, let's calculate the final energy:

The final kinetic energy is given by:

KE(final) = 0.5 * mass * (velocity)^2

Substituting the given values, we get:

KE(final) = 0.5 * 0.1 kg * (5 m/s)^2 = 1.25 J

At a height of 4.7 m, the potential energy can be calculated as:

PE(final) = mass * gravity * height

Substituting the given values, where gravity is approximately 9.8 m/s^2, we get:

PE(final) = 0.1 kg * 9.8 m/s^2 * 4.7 m = 4.586 J (approx)

Therefore, the final energy is:

Final Energy = KE(final) + PE(final) = 1.25 J + 4.586 J = 5.836 J (approx)

To find the fraction of energy lost to air friction, we need to find the difference between the initial and final energies:

Energy Lost = Initial Energy - Final Energy

Energy Lost = 12.8 J - 5.836 J = 6.964 J (approx)

Now, let's calculate the fraction of energy lost by dividing the energy lost by the initial energy:

Fraction of Energy Lost = Energy Lost / Initial Energy

Fraction of Energy Lost = 6.964 J / 12.8 J ≈ 0.543

Therefore, approximately 54.3% of the ball's original energy has been lost to air friction.