A brother and sister are standing next to each other at rest on a surface of frictionless ice. The brother’s mass is exactly twice that of his sister’s. The sister suddenly pushes her brother. As a result, the sister moves with kinetic energy K. What is the resulting kinetic energy of the brother?
i know the answer is K/2 but i don't know why. Can someone explain it to me please?
Ok so I struggled with this too, but I think I just figured out how to do it.
You can think of this problem as an elastic collision(hopefully) where Ke is conserved and momentum is conserved. I'm not going to bring momentum into this, but. Kbrother = Ksister
Ksister = 1/2 * M * V^2
Kbrother = 1/2 * 2M * V^2
We can break this down into
Kbrother = 2 * (1/2MV^2)
Ksister = (1/2MV^2)
Kbrother = 2 * (Ksister)
Kbrother/2 = Ksister
so that's how I did it and thought about it. I don't know if that's the right way to do it though.
Well, let me put it in a funny way for you!
Imagine the brother and sister are standing on the ice, minding their own business. The sister, filled with mischief, decides to give her brother a little push. Now, let's see what happens!
When the sister pushes her brother, they both start moving in opposite directions. It's like a comedy show where they slip on a banana peel! Of course, since the sister is smaller, she moves faster than her brother. She's like Usain Bolt, zooming away!
But here's the trick: remember that kinetic energy depends on both mass and speed. The sister zooms away with a certain speed, let's call it "S".
Since kinetic energy is proportional to the square of the speed, her kinetic energy is K = mS^2, where "m" represents the sister's mass.
Now, let's look at the poor brother. He's heavier, so he doesn't move as quickly as his sister. He's more like a turtle running a marathon! His speed, let's call it "B", is slower than his sister's speed.
But here's the punchline: the brother has double the mass of his sister! So while his speed is lower, his mass evens things out. His kinetic energy is K' = (2m)(B^2), where "m" represents the brother's mass.
Now, let's compare their kinetic energies. We have K = mS^2 for the sister and K' = (2m)(B^2) for the brother.
Since we're given that K = K/2, we can equate the two expressions:
K = K'
mS^2 = (2m)(B^2)
S^2 = 2B^2
And that's how we find out that the sister's kinetic energy is indeed K, while the brother's kinetic energy is half of that, K/2!
In conclusion, the sister went full-speed ahead while the brother braked his way to a slower-but-still-energetic motion, resulting in him having half the kinetic energy of his sister. Brothers, am I right? Always getting the sibling discount!
I hope that puts a smile on your face and clarifies why the resulting kinetic energy of the brother is K/2. Keep those physics questions coming!
In this scenario, we can analyze the situation using the principles of conservation of momentum and conservation of kinetic energy.
According to the law of conservation of momentum, the total momentum before the push is equal to the total momentum after the push. Since the brother and sister were initially at rest, their total momentum before the push is zero.
After the sister pushes her brother, both the brother and the sister will start moving in opposite directions. Let the brother's velocity immediately after the push be Vb and the sister's velocity be Vs.
Since the sister pushes the brother, the sister's velocity after the push will be greater than zero (Vs > 0), and the brother's velocity will be less than zero (Vb < 0). This is due to the conservation of momentum.
By definition, kinetic energy is given by the equation K = (1/2)mv^2, where m is the mass and v is the velocity.
Since the sister pushes the brother, the brother gains velocity and hence kinetic energy. However, the sister loses velocity and thus kinetic energy. But since the total kinetic energy must be conserved, the decrease in the sister's kinetic energy must be equal to the increase in the brother's kinetic energy.
Since the brother's mass is exactly twice that of the sister's and the equation for kinetic energy is proportional to the mass, the brother will have half the velocity (Vb = -Vs/2) and hence half the kinetic energy of the sister (Kb = K/2).
Therefore, the resulting kinetic energy of the brother is K/2.
To explain why the resulting kinetic energy of the brother is K/2, let's understand the concept of conservation of momentum and energy in this scenario.
In this situation, both the brother and sister are initially at rest, which means they have zero kinetic energy. When the sister pushes her brother, she imparts an equal and opposite force on him. According to Newton's third law of motion, for every action, there is an equal and opposite reaction.
As a result of the push, the sister starts moving, acquiring kinetic energy denoted by K. At the same time, the brother also starts moving, but with a different velocity than the sister due to their mass difference.
According to the law of conservation of momentum, the total momentum before and after the sister pushes her brother should be equal. Since the total momentum before the push is zero (both standing still), the total momentum after the push should also be zero.
Now, let's calculate the resulting kinetic energy of the brother:
Let M_b be the mass of the brother and M_s be the mass of the sister.
Since the brother's mass is exactly twice that of the sister’s (M_b = 2M_s), the sister's velocity after the push will be twice that of the brother's velocity.
Let v_s and v_b be the velocities of the sister and brother after the push, respectively.
The total momentum after the push is given by:
Total Momentum = (Mass of Sister * Velocity of Sister) + (Mass of Brother * Velocity of Brother)
= (M_s * v_s) + (M_b * v_b)
Since the total momentum before and after the push should be equal (zero in this case), we have:
(M_s * v_s) + (M_b * v_b) = 0
Now, let's consider the kinetic energies of the sister and brother:
Kinetic energy is given by the equation: Kinetic Energy = 1/2 * mass * velocity^2
The kinetic energy of the sister, K, is given by:
K = 1/2 * M_s * v_s^2
The kinetic energy of the brother, let's call it K_b, is given by:
K_b = 1/2 * M_b * v_b^2
Since the sister's velocity is twice that of the brother (v_s = 2 * v_b), we can substitute this into the kinetic energy equations:
K = 1/2 * M_s * (2 * v_b)^2
= 1/2 * M_s * 4 * v_b^2
= 2 * 1/2 * M_s * v_b^2
= 2 * K_b
Therefore, the resulting kinetic energy of the brother is K/2.