Why Can't We Calculate the Height of a Ball Thrown Up Without Air Resistance?

In summary: Additionally, the ball will eventually reach a point where it will decelerate to 0, and at that point the air resistance will be zero. So you can ignore the last two equations in the problem and simply say that the ball will travel a distance equal to the work done by air resistance, or 40 m.
  • #1
elissadi
10
0
If a 3 kg ball is thrown straight up at 40 m/s, using energy conservation, calculate how high the ball would go if there was no wind resistance.

I was told this could not be calculated. Is that true? and if so, why?
 
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  • #2
why not, loss in KE = gain in PE ?
 
  • #3
What is the formula?
What if I added, let's say, a max. height like 75 m. Could energy lost, like due to friction with the air on the way up be calculated?
 
  • #4
This can be done by looking at a symmetric problem. Assume the ball does decelerate to 0. Then by symmetry, solve the problem of how far it would have had to accelerate under gravity near the Earth's surface to get to 40 m/s, since assuming it loses no energy to air resistance this is the velocity it would return with to the person's hand.
If you want to add in air resistance, you have to account for the cross-sectional area of the ball (A), as a sheet of paper will fall slower than a crumpled up piece. Similarly, a styrofoam ball will encounter more resistance than a rubber ball, so density (d) is inversely proportional to the air resistance. Next, stick your hand out when you're walking. Compare it to sticking your hand out when you're running or in a car. Speed (v) is thus proportional to air resistance. We're insterested in the distance the ball will travel. m is the mass of the ball. So far we have the rudimentary 1-dimensional net force equation ma = mg - A*d*v. But v is not a constant, it is a function of time, so we have to solve the differential equation m*dv/dt = - mg - A*d*v for speed v or m*d2x/dt2 = - mg - A*d*dx/dt for distance x. This simple model has many more refinements placed on it based on the speed of the object and the type of resistance, but solving it will already allow you to see simple behavior like terminal velocity. :smile:
Since the energy lost by the ball is equivalent to the work done by the air resistance on the ball, you want to calculate the total work done by air resistance in the above equations, which would require integration, since the air resistance is also a function of time.
 
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1. What is the law of conservation of energy?

The law of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. This means that the total amount of energy in a closed system remains constant over time.

2. Why is the conservation of energy important?

The conservation of energy is important because it is a fundamental law of nature that governs the behavior of all physical systems. It allows us to understand and predict how energy is transferred and transformed in different processes, from microscopic interactions to global energy systems.

3. What are some examples of energy conservation in everyday life?

Examples of energy conservation in everyday life include turning off lights when leaving a room, using energy-efficient appliances, and carpooling to reduce fuel consumption. Conserving energy not only saves money, but also helps to reduce our overall energy consumption and environmental impact.

4. How does the conservation of energy relate to the concept of efficiency?

The conservation of energy is closely related to the concept of efficiency. Efficiency measures how well a system converts energy from one form to another. The more efficient a system is, the less energy is wasted and the closer it is to achieving 100% conservation of energy.

5. Can energy be lost or disappear in a system?

No, according to the law of conservation of energy, energy cannot be lost or disappear in a system. It can only be transformed into different forms, such as heat, light, or mechanical energy. This means that the total amount of energy in a system will always remain constant, even if it changes forms.

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