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julsineya [31]
3 years ago
11

Una pelota de basket es soltada desde 2.5 m de altura y rebota con una velocidad igual a 3/4 partes de la velocidad que llego. ¿

a qué altura alcanza la bola en el rebote ? ¿ cuánto tiempo transcurre desde que rebota ?
Physics
1 answer:
FrozenT [24]3 years ago
4 0

Answer:

Tenemos dos problemas a resolver acá:

Primero, debemos encontrar la velocidad con la que la pelota impacta el suelo.

Acá podemos usar la conservación de la energía.

E = U + K

U = energía potencial = m*g*H

m = masa

g = aceleración gravitatoria = 9.8m/s^2

H = altura

K = energía cinética = (m/2)*V^2

donde V es la velocidad.

Inicialmente, cuando la pelota es soltada, su velocidad es cero, entonces solo tenemos energía potencial:

Ei = U = m*(9.8m/s^2)*2.5m

Al final, cuando la pelota esta por impactar el suelo, la altura tiende a cero, entonces ya no hay energía potencial, solo hay energía cinética:

Ef = (m/2)*V^2

Y como la energía se conserva, la energía final es igual a la inicial, entonces:

m*(9.8m/s^2)*2.5m = (m/2)*V^2

Podemos resolver esto para V, y asi obtener la velocidad con la que la pelota impacta el suelo.

V = √(2*(9.8m/s^2)*2.5m) = 7m/s

Ahora respondamos la segunda parte.

Una vez la pelota rebota, su aceleración va a estar dada solamente por la aceleración gravitatoria, entonces tenemos:

A(t) = -9.8m/s^2

Para obtener su velocidad integramos:

V(t) = (-9.8m/s^2)*t + V0

donde V0 es la velocidad con la que la pelota reboto, que sabemos que es 3/4 de 7m/s

V0 = (3/4)*7m/s = (21/4) m/s

Así, la ecuación de la velocidad es:

V(t) = (-9.8m/s^2)*t + (21/4) m/s

Sabemos que la altura máxima se da cuando la velocidad es igual a cero, entonces primero calculemos el valor de t tal que esto ocurra:

V(t) = 0 = (-9.8m/s^2)*t + (21/4) m/s

         t =  (21/4) m/s/9.8m/s^2 = 0.54 s

Ahora debemos encontrar la ecuación de la posición y evaluarlo en este tiempo.

Para ello integramos de vuelta:

P(t) = (1/2)(-9.8m/s^2)*t^2 + (21/4 m/s)*t + P0

donde P0 es la posición inicial, como la pelota rebota en el suelo, la posición inicial es el suelo, el cual representamos con 0, entonces la ecuación de la posición es:

P(t) = (1/2)(-9.8m/s^2)*t^2 + (21/4 m/s)*t  

La altura máxima estará dada por esta ecuación evaluada en t = 0.54 s

P(0.54s) =  (1/2)(-9.8m/s^2)*(0.54s)^2 + (21/4 m/s)*0.54s = 1.81 m

La altura máxima es 1.81 metros.

Y entre que rebota y llega a esta altura máxima, transcurren 0.54 segundos.

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