In a Broadway performance, an 80 kg actor swings from a 3.75 m long cable that is horizontal when he starts.?
At the bottom of his arc, he picks up his 55.0 kg costar in an inelastic collision. 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(2gL)=...
(b) Law of conservation of linear momentum
mv= (m+m1)u
u= m•v/(m+m1)=...
(c)
Law of conservation of energy
KE1=PE1
(m+m1) •u²/2 =(m+m1) •g•h1
h1=(m+m1) •u²/2 • (m+m1) •g =...
KE1=PE1
in the end should actually be...
h1=(m+m1)(u^2)/2*g*(m+m1)
where the sum of the masses cancel and you're left with: (u^2)/2*g
To find the maximum height reached by the actors after their upward swing, we can use the principle of conservation of mechanical energy.
The initial energy of the system (actor swinging alone) is equal to the final energy of the system (actors at maximum height).
1. Calculate the initial potential energy (PI) of the system when the actor is swinging alone:
PI = m1 * g * h1
Where:
m1 = mass of the swinging actor = 80 kg
g = acceleration due to gravity = 9.8 m/s^2
h1 = initial height from the ground = length of the cable = 3.75 m
PI = 80 kg * 9.8 m/s^2 * 3.75 m
2. Calculate the initial kinetic energy (KI) of the system when the actor is swinging alone:
KI = 0.5 * m1 * v1^2
Where:
v1 = velocity at the bottom of the swing
Since the actor starts from rest at the bottom of the swing, v1 = 0.
KI = 0.5 * 80 kg * (0 m/s)^2 = 0 J (Joules)
3. Calculate the initial total mechanical energy (ETI) of the system when the actor is swinging alone:
ETI = PI + KI = PI (since KI = 0 in this case)
ETI = 80 kg * 9.8 m/s^2 * 3.75 m
4. Calculate the final potential energy (PF) of the system when both actors reach maximum height:
PF = (m1 + m2) * g * h2
Where:
m2 = mass of the costar = 55 kg
h2 = maximum height reached by the actors
To find h2, we need to determine the final velocity (vf) of the system after the collision.
5. Use the conservation of linear momentum to determine the final velocity (vf) of the system:
(m1 * v1) + (m2 * 0) = (m1 + m2) * vf
Where:
v1 = initial velocity of the actor
Since the actor starts from rest at the bottom of the swing, v1 = 0.
(80 kg * 0 m/s) + (55 kg * 0) = (80 kg + 55 kg) * vf
0 kg·m/s + 0 kg·m/s = 135 kg * vf
vf = 0 m/s
The final velocity of the system after the collision is 0 m/s since both actors momentarily come to rest at the top of the swing.
6. Calculate the final kinetic energy (KF) of the system when both actors reach maximum height:
KF = 0.5 * (m1 + m2) * vf^2
KF = 0.5 * (80 kg + 55 kg) * (0 m/s)^2 = 0 J (Joules)
7. Calculate the final total mechanical energy (ETF) of the system when both actors reach maximum height:
ETF = PF + KF
Since KF = 0, ETF = PF
ETF = (m1 + m2) * g * h2
8. Equate the initial and final mechanical energies to find the maximum height (h2):
ETI = ETF
Therefore, 80 kg * 9.8 m/s^2 * 3.75 m = (80 kg + 55 kg) * 9.8 m/s^2 * h2
Simplifying the equation:
Great! We solve for h2:
h2 = (80 kg * 9.8 m/s^2 * 3.75 m) / ((80 kg + 55 kg) * 9.8 m/s^2)
h2 = (3132 J) / (135 kg * 9.8 m/s^2)
h2 ≈ 2.2 m
Therefore, the maximum height reached by the actors after their upward swing is approximately 2.2 meters.
To find the maximum height reached after the upward swing, we need to determine the initial speed of the actors as they start swinging. Then we can calculate the potential energy at the maximum height using the conservation of energy principle.
1. Find the initial speed:
Since the cable is horizontal when the actor starts swinging, there is no gravitational potential energy initially. Therefore, the total mechanical energy is equal to the kinetic energy (KE) at that point.
The formula for kinetic energy is:
KE = 0.5 * mass * velocity^2
Given:
Actor's mass (m1) = 80 kg
Costar's mass (m2) = 55 kg
Cable length (L) = 3.75 m
Let's calculate the initial velocity with which the actor starts swinging:
Since the cable length and the horizontal position is given, we can create an equation using the conservation of energy at the starting position, setting it equal to the kinetic energy.
Also, we know that the only forces acting on the actor at the lowest point of the swing are the actor's weight and the tension in the cable. These forces are perpendicular, so their vector sum must give the centripetal force needed for circular motion.
Hence, we can use the following equation which relates the gravitational force, tension force, and centripetal force:
m1 * g * sinθ = m1 * (v0^2 / L)
As sinθ is equal to 1 (since the cable is horizontal), we can simplify the equation to:
g = (v0^2 / L)
Rearranging the equation for initial velocity:
v0 = sqrt(g * L)
Now, we need to calculate the value of acceleration due to gravity (g), which is approximately 9.8 m/s^2:
v0 = sqrt(9.8 m/s^2 * 3.75 m)
v0 ≈ 7.22 m/s
2. Calculate the potential energy at the maximum height:
At the maximum height, all of the initial kinetic energy is converted into potential energy (PE). The sum of the potential energy and the kinetic energy will be equal to the initial kinetic energy.
So, the equation becomes:
KE_initial = PE_max + KE_max
Rearranging and replacing the values:
PE_max = KE_initial - KE_max
Given:
Initial total kinetic energy (KE_initial) = KE_actor_initial = 0.5 * m1 * v0^2
Final total kinetic energy (KE_max) = KE_actor_max + KE_costar_max
KE_actor_max = 0.5 * m1 * vf^2
KE_costar_max = 0.5 * m2 * vf^2
where vf is the final velocity at the maximum height
Since it's an inelastic collision, the final velocity vf of both the actor and the costar will be the same. Let's denote it as v_max.
Now, substitute the values and solve for PE_max:
PE_max = 0.5 * m1 * v0^2 - (0.5 * m1 * vf^2 + 0.5 * m2 * vf^2)
Now we have v0 and vf. Let's calculate PE_max.
PE_max = 0.5 * 80 kg * (7.22 m/s)^2 - (0.5 * 80 kg * vf^2 + 0.5 * 55 kg * vf^2)
3. Calculate the maximum height:
Since potential energy is given by the formula PE = m * g * h, where m is the total mass and g is the acceleration due to gravity, we can solve for h (the maximum height).
Using the formula:
PE_max = m * g * h,
we can rearrange it:
h = PE_max / (m * g)
Let's calculate the maximum height:
h = PE_max / ((m1 + m2) * g)
h = (PE_max) / ((80 kg + 55 kg) * 9.8 m/s^2)
By substituting the value of PE_max into the equation, the calculation will yield the answer for the maximum height reached after the upward swing.