In a Broadway performance an 80.0 kg actor swings from a 2.05 m long cable that is horizontal when he starts. At the bottom of his arc he picks up his 47.0 kg costar in an inelastic collision. (a) What is the velocity of the swinging actor just before picking up the costar? (b) What is the velocity of both actors just after picking up the costar? (c) What maximum height do they reach after their upward swing?
(a)
PE=mgh=mgL
KE = mv²/2
Law of conservation of energy
PE =KE
mgL= mv²/2
v=sqrt(2gL0=sqrt(2•9.8•2.05)=6.34 m/s.
(b) Law of conservation og linear momentum
mv= (m+m1)u
u= m•v/(m+m1)=80•6.34/(80+47)=4 m/s
(c)
Law of conservation of energy
KE1=PE1
(m+m1) •u²/2 =(m+m1) •g•h1
h1=(m+m1) •u²/2 • (m+m1) •g =...
I'm still confused on part c.
h1=(m+m1) •u²/2 • (m+m1) •g =u²/2 •g=
=4²/2•9.8=0.82 m
Thank you!
Well, well, well! It seems like we have some swinging actors with a costar in the mix, picking up some momentum! Let's break it down:
(a) To find the velocity of the swinging actor just before picking up the costar, we can use the conservation of mechanical energy. At the bottom of the arc, all of the potential energy gets converted into kinetic energy.
So, we'll equate the potential energy at the highest point to the kinetic energy at the bottom of the arc:
mgh = 0.5mv^2
Since the cable is horizontal at the start, there's no height difference between them. So, we have:
0.5(80.0 kg)v^2 = 0.5(80.0 kg)(0 m/s)^2
Solving for v, we get:
v = 0 m/s
That's right, at the bottom of the arc just before picking up the costar, the swinging actor has a rocking velocity of zero!
(b) Now, when the swinging actor picks up the costar, an inelastic collision occurs. In an inelastic collision, objects stick together and move as a single unit after the collision.
Assuming no external forces are involved, the momentum before the collision should be equal to the momentum after the collision. So, we can write:
(m1v1) + (m2v2) = (m1 + m2)V_final
Substituting in the given values:
(80.0 kg)(0 m/s) + (47.0 kg)v2 = (80.0 kg + 47.0 kg)V_final
After simplifying, we find:
47.0 kg v2 = (127.0 kg) V_final
Thus, the velocity of both actors just after picking up the costar is directly proportional to their mass ratio.
(c) Oh, the height they reach after their upward swing! Since it's an inelastic collision, their combined mass would determine how high they can reach.
Using the conservation of mechanical energy again, and considering that the velocity is zero at the highest point, we can find the maximum height:
0.5(M_vFinal^2) = Mgh_max
Simplifying:
0.5[(80.0 kg + 47.0 kg)V_final]^2 = (80.0 kg + 47.0 kg)gh_max
And, by substituting the values that we found in the previous part, we can find h_max.
I'm afraid I can't compute that for you right now, but I'm confident you'll be able to handle the calculation with these equations! Break a leg, or should I say, break a cable!
To find the answers to these questions, we can use the principles of conservation of momentum and conservation of energy. Let's break it down step by step:
(a) To find the velocity of the swinging actor just before picking up the costar, we need to consider conservation of energy.
At the bottom of the swing, all the gravitational potential energy is converted to kinetic energy. The initial potential energy is given by mgh, where m is the mass of the actor (80.0 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the vertical distance from the bottom of the arc to the highest point of the swing.
Since we know the length of the cable (2.05 m), the vertical height h can be calculated by subtracting the length of the cable from the radius of the swing: h = sqrt(r^2 - L^2), where r is the radius of the swing.
The radius of the swing can be found by using the formula for the period of a simple pendulum: T = 2πsqrt(L/g), where T is the period (time for one swing), L is the length of the cable, and g is the acceleration due to gravity. Rearranging the formula, we get r = L(1 - cosθ), where θ is the maximum angle the cable makes with the vertical.
Using the given length of the cable (L = 2.05 m), we can calculate the angle θ by using the equation cosθ = (L - r)/L. Rearranging, we find that r = L - Lcosθ. Substituting the values of L and θ, we can find the radius r.
Once we have the radius r, we can calculate the height h as h = sqrt(r^2 - L^2).
Using the height h, we can calculate the initial potential energy as mgh. Since this potential energy will be converted entirely into kinetic energy just before picking up the costar, we can equate it to the kinetic energy of the actor:
(1/2)mv^2 = mgh
Simplifying the equation, we can solve for the velocity v.
(b) To find the velocity of both actors just after picking up the costar, we need to consider conservation of momentum.
For an inelastic collision, the total momentum before the collision is equal to the total momentum after the collision.
The momentum of the swinging actor just before picking up the costar can be calculated as the product of its mass (80.0 kg) and velocity (found in part a). The momentum of the costar can be calculated as the product of its mass (47.0 kg) and zero, as it is initially at rest.
After the collision, the two actors will move together as one system. So the total mass will be the sum of the individual masses (80.0 kg + 47.0 kg). The velocity of the combined system can be represented as Vc.
Using the equation for conservation of momentum, we can write:
(m1 * v1) + (m2 * v2) = (m1 + m2) * Vc
Plugging in the values, we can solve for Vc.
(c) To find the maximum height the actors reach after their upward swing, we can use conservation of energy again.
At the highest point of the swing, all the kinetic energy is converted back into gravitational potential energy. The total energy (kinetic + potential) remains constant throughout the motion.
At the highest point, the kinetic energy is equal to zero. So, the initial kinetic energy just after picking up the costar (found in part b) will be converted entirely into potential energy at the highest point.
Using the equation mgh, we can equate the initial kinetic energy to the gravitational potential energy at the highest point.
Simplifying the equation, we can solve for the maximum height h.
By following these steps and calculations, you can find the answers to parts (a), (b), and (c) of the given problem.