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marusya05 [52]
2 years ago
9

A 55.6-kg skateboarder starts out with a speed of 2.44 m/s. He does 80.4 J of work on himself by pushing with his feet against t

he ground. In addition, friction does -244 J of work on him. In both cases, the forces doing the work are non-conservative. The final speed of the skateboarder is 7.24 m/s. (a) Calculate the change (PEf - PE0) in the gravitational potential energy. (b) How much has the vertical height of the skater changed? Give the absolute value.
Physics
1 answer:
Vikentia [17]2 years ago
7 0

(a) -1620.8 J

The initial kinetic energy of the  skateboarder is:

K_i = \frac{1}{2}mu^2 = \frac{1}{2}(55.6 kg)(2.44 m/s)^2=165.5 J

where m is the skateboarder's mass and u his initial speed;

While the final kinetic energy is

K_f = \frac{1}{2}mv^2 = \frac{1}{2}(55.6 kg)(7.24 m/s)^2=1457.2 J

where v is his final speed.

So the change in kinetic energy is

\Delta K=K_f - K_i = 1457.2 J -165.6 J = 1291.6 J

According to the work-energy theorem, the change in mechanical energy (kinetic+potential) of the skateboarder is equal to the work done on it:

\Delta K + \Delta E_p = W + W_f

where

W = 80.4 J is the work done by the skateboarder on himself

W_f = -244 J is the work done by friction

\Delta E_p = E_p_f - E_p_i is the change in gravitational potential energy

Solving for \Delta E_p,

\Delta E_p = W+W_f - \Delta K=80.4 J - 244 J - 1457.2 J = -1620.8 J

(b) 2.97 m

The change in potential energy of the skateboarder can be written as

\Delta E_p = mg \Delta h

where

m = 55.6 kg is the mass

g = 9.8 m/s^2 is the acceleration of gravity

\Delta h is the change in vertical height of the skateboarder

Solving for \Delta h,

\Delta h = \frac{\Delta E_p}{mg}=\frac{-1620.8 J}{(55.6 kg)(9.8 m/s^2)}=-2.97 m

Where the negative sign means the skateboarder has moved downwards. Since we are interested only in the absolute value, the answer is

h = 2.97 m

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When the spring is compressed downward, the spring has elastic potential energy. When the spring is relaxed, there is no elastic potential energy, but the pellet will have gained gravitational potential energy and kinetic energy. Lets see what are the terms for each of this.

<h3>Elastic potential energy</h3>

We know that a spring following Hooke's Law has a elastic potential energy:

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To obtain the spring's constant, we know that Hooke's law states that the force made by the spring is :

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as we need 9.12 N to compress 4.60 cm, this means:

k = \frac{9.12 \ N}{4.6 \ 10^{-2} \ m}

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E_{ep} = \frac{1}{2} 198.26 \ \frac{ N}{m} (4.6 \ 10^{-2} \ m)^2

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To see how much gravitational potential energy will the pellet win, we can use

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Taking all this together, the gravitational potential energy when the spring is relaxed will be:

\Delta E_{gp} = 4.97 \ 10^{-3} kg \ 9.8 \frac{m}{s^2} 4.6 \ 10^{-2} m

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We know that the kinetic energy for a mass m moving at speed v is:

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E_k = \frac{1}{2} \ 4.97 \ 10^{-3} kg \ v^2

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By conservation of energy, we know:

E_{ep} = \Delta E_{gp} + E_k

0.209759 \ Joules = 0.00224 \ Joules + \frac{1}{2} \ 4.97 \ 10^{-3} kg \ v^2

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