Mechanical Energy: Trampoline Artist Jumps and Depresses Spring

In summary, mechanical energy is the energy possessed by an object due to its motion or position. It can be categorized into two types: kinetic energy, which is the energy an object has due to its motion, and potential energy, which is the energy an object has due to its position or shape. In the context of trampoline jumping, mechanical energy is the energy that is transferred between the trampoline and the jumper. The spring plays a crucial role in the mechanical energy of a trampoline artist, as it stores potential energy when compressed and converts it into kinetic energy when decompressed. The height of the jump directly affects the mechanical energy, with a higher jump resulting in a more powerful display of mechanical energy. However, the weight of the tr
  • #1
rphmy
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0
A 75-kg trampoline artist jumps vertically upward from the top of a platform with a speed of 5.0 m/s. (a) how fast is he going as he lands on the trampoline 3.0 m below (Fig. 6-23)? (b) If the trampoline behaves like a spring of spring constant 5.2x10^4 N/m, how far does he depress it?
 
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I think this is a kind of homework
 
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a) To calculate the speed of the trampoline artist as he lands on the trampoline, we can use the conservation of mechanical energy principle. Since the artist is initially at rest at the top of the platform, all of his mechanical energy is in the form of potential energy. As he jumps, this potential energy is converted into kinetic energy. At the bottom of the jump, all of his initial potential energy will be converted into kinetic energy again. Therefore, we can use the equation for conservation of mechanical energy:

Initial potential energy = Final kinetic energy

mgh = (1/2)mv^2

Where m is the mass of the artist, g is the acceleration due to gravity (9.8 m/s^2), h is the height of the platform (3.0 m), and v is the final velocity.

Solving for v, we get:

v = √(2gh)

Substituting the given values, we get:

v = √(2 x 9.8 m/s^2 x 3.0 m) = 7.66 m/s

Therefore, the trampoline artist will be going at a speed of 7.66 m/s as he lands on the trampoline.

b) To calculate the distance the trampoline will depress, we can use Hooke's law, which states that the force exerted by a spring is directly proportional to the distance it is stretched or compressed. We can use the equation:

F = kx

Where F is the force exerted by the spring, k is the spring constant (5.2x10^4 N/m), and x is the distance the spring is compressed.

We know that the force exerted by the spring is equal to the weight of the trampoline artist, which is given by:

F = mg = (75 kg) x (9.8 m/s^2) = 735 N

Substituting this into the equation, we get:

735 N = (5.2x10^4 N/m) x x

Solving for x, we get:

x = 735 N / (5.2x10^4 N/m) = 0.0141 m

Therefore, the trampoline will depress by 0.0141 m when the artist lands on it.
 

1. What is mechanical energy?

Mechanical energy is the energy possessed by an object due to its motion or position. It can be categorized into two types: kinetic energy, which is the energy an object has due to its motion, and potential energy, which is the energy an object has due to its position or shape.

2. How is mechanical energy related to trampoline jumping?

In the context of trampoline jumping, mechanical energy is the energy that is transferred between the trampoline and the jumper. When the jumper jumps on the trampoline, they compress the springs, which stores potential energy. As the springs decompress, this potential energy is converted into kinetic energy, propelling the jumper upwards.

3. How does the spring affect the mechanical energy of the trampoline artist?

The spring plays a crucial role in the mechanical energy of a trampoline artist. When the artist jumps on the trampoline, the spring compresses, storing potential energy. As the artist continues to jump and the spring decompresses, this potential energy is converted into kinetic energy, allowing the artist to jump higher.

4. How does the height of the jump affect the mechanical energy of the trampoline artist?

The height of the jump directly affects the mechanical energy of the trampoline artist. The higher the jump, the more potential energy is stored in the spring and the more kinetic energy is produced when the spring decompresses. This results in a more powerful jump and a higher display of mechanical energy.

5. How does the weight of the trampoline artist affect the mechanical energy of the jump?

The weight of the trampoline artist does not have a significant effect on the mechanical energy of the jump. This is because the potential energy stored in the spring is determined by the amount of compression, not the weight of the jumper. However, a heavier jumper may need to exert more force to compress the spring, resulting in a slightly higher display of mechanical energy.

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