You see a pair of shoes tied together by the laces and hanging over a telephone line. You throw a 0.21-kg stone at one of the shoes (mass = 0.35 kg), and it collides elastically with the shoe with a velocity of 1.8 m/s in the horizontal direction. How far up does the shoe move?
Assume the rock hits a single shoe side-on and that kinetic energy and momentum are conserved on impact. The mass of a single shoe is 0.175 kg. Since that is approximately equal to the stone's mass, the shoe will acquire a velocity approximately equal to the stone's velocity before collision.
(You should work that out on your own).
From the velocity of the single shoe after impact (about 1.8 m/s), calculate how far it will rise while hanging by a shoestring.
To find the distance the shoe moves up after the collision, we need to use the principle of conservation of momentum. Here are the steps to determine the distance.
Step 1: Calculate the initial momentum of the stone and the shoe before the collision.
The formula for momentum is:
Momentum (p) = mass (m) x velocity (v)
The momentum of the stone before the collision is:
Momentum of stone = mass of stone x velocity of stone = 0.21 kg x 1.8 m/s
The momentum of the shoe before the collision is:
Momentum of shoe = mass of shoe x velocity of the shoe = 0.35 kg x 0 m/s (since it's at rest)
Step 2: Calculate the total momentum after the collision.
Since the collision is elastic, the total momentum before the collision should be equal to the total momentum after the collision.
So, the total momentum after the collision is:
Total momentum = momentum of stone + momentum of shoe
Step 3: Calculate the velocity of the shoe after the collision.
Since we know the mass and the total momentum, we can calculate the velocity of the shoe after the collision using the formula for momentum.
Total momentum = mass of shoe x velocity of shoe
Solving for velocity of shoe: velocity of shoe = total momentum / mass of shoe
Step 4: Calculate the distance the shoe moves up.
To find the distance the shoe moves up, we can use the formula for the vertical motion of an object under the influence of gravity.
Vertical displacement (d) = (velocity^2) / (2 x gravity)
Since the initial velocity of the shoe is 0 m/s (at rest) and we know the velocity of the shoe after the collision from step 3, we can calculate the vertical displacement.
Putting it all together:
1. Calculate the initial momentum:
Momentum of stone = 0.21 kg x 1.8 m/s
2. Calculate the total momentum after the collision:
Total momentum = Momentum of stone + momentum of shoe
3. Calculate the velocity of the shoe after the collision:
Velocity of shoe = Total momentum / mass of shoe
4. Calculate the vertical displacement:
Vertical displacement = (Velocity of shoe^2) / (2 x gravity)
Note: The value of gravity can be taken as 9.8 m/s^2.
Using these steps, you can find the distance the shoe moves up after the collision.
To determine how far the shoe moves up, we can analyze the conservation of momentum and energy in this elastic collision problem.
First, let's calculate the initial momentum of the stone and the shoe before the collision. The momentum is given by the equation:
p = m * v
Where p is the momentum, m is the mass, and v is the velocity.
The momentum of the stone before the collision is:
momentum_stone_initial = mass_stone * velocity_stone_initial
momentum_stone_initial = 0.21 kg * 1.8 m/s
The momentum of the shoe before the collision is:
momentum_shoe_initial = mass_shoe * velocity_shoe_initial
momentum_shoe_initial = 0.35 kg * 0 m/s (since the shoe is initially at rest)
Since the collision is elastic, the total momentum before and after the collision should be conserved. So the total momentum before the collision is equal to the total momentum after the collision:
momentum_total_initial = momentum_stone_initial + momentum_shoe_initial
momentum_total_initial = 0.21 kg * 1.8 m/s + 0.35 kg * 0 m/s
Next, let's calculate the final velocity of the stone and the shoe after the collision. Since the collision is elastic, both the stone and the shoe will move after the collision.
We can use the law of conservation of momentum to set up an equation:
momentum_total_initial = momentum_stone_final + momentum_shoe_final
momentum_stone_final = (momentum_total_initial - momentum_shoe_final)
Now, let's consider the conservation of energy. Since this is an elastic collision, both momentum and kinetic energy are conserved. The initial kinetic energy is equal to the sum of the final kinetic energies of the stone and the shoe:
kinetic_energy_initial = kinetic_energy_stone_final + kinetic_energy_shoe_final
Using the equation for kinetic energy:
kinetic_energy = (1/2) * mass * velocity^2
The initial kinetic energy is given by:
kinetic_energy_initial = (1/2) * mass_stone * velocity_stone_initial^2
The final kinetic energy of the stone is:
kinetic_energy_stone_final = (1/2) * mass_stone * velocity_stone_final^2
Similarly, the final kinetic energy of the shoe is:
kinetic_energy_shoe_final = (1/2) * mass_shoe * velocity_shoe_final^2
Since the collision is elastic, the total kinetic energy before the collision should be equal to the total kinetic energy after the collision. So, we can set up an equation:
kinetic_energy_initial = kinetic_energy_stone_final + kinetic_energy_shoe_final
Lastly, we need to consider the gravitational potential energy of the shoe when it moves up. The change in gravitational potential energy can be calculated using the formula:
change_in_potential_energy = mass * gravity * height
Where mass is the mass of the shoe, gravity is the acceleration due to gravity, and height is the height the shoe moves up.
We can now use the above equations and principles to solve for the height the shoe moves up.