a space shuttle astronaut has a mass of 110 kg with her spacesuit on. She is on a spacewalk and picks up a full can of spray with a mass of 20.8kg. Relative to the space shuttle, she is at rest. She then holds the can directly in front of her center of mass to avoid rotation and releases 5.0kg of spray at a speed of 16.4m/s. Her speed, relative to the space shuttle, when she has stopped spraying is approximately a.bc m/s.
Well, I don't know about you, but I'm just glad she didn't spray the can directly into her face! That could've been a real "space-tastrophe"!
But let's get down to business. To solve this, we can use the principle of conservation of momentum. The initial momentum of the astronaut and the can must be equal to the final momentum when she stops spraying.
The momentum of an object is given by the formula: momentum = mass x velocity.
So, the initial momentum is the sum of the astronaut's momentum and the can's momentum. Since she's at rest relative to the space shuttle initially, her momentum is 0 kg*m/s.
The momentum of the can is its mass (20.8 kg) multiplied by its initial velocity (which we'll call v_init). The final momentum of the combined system (astronaut + can) is the sum of their individual momenta.
When she stops spraying, the momentum of the can will be 0 because it's not moving anymore. So, we can set up an equation:
0 kg*m/s = (110 kg x v_astronaut) + (20.8 kg x v_init)
Now we just need to rearrange the equation to solve for v_astronaut (her velocity relative to the space shuttle when she stops spraying).
v_astronaut = - (20.8 kg x v_init) / 110 kg
Plugging in the given values, we get:
v_astronaut = - (20.8 kg x 16.4 m/s) / 110 kg
Doing the math, we find that v_astronaut is approximately -3.08 m/s.
So, her speed, relative to the space shuttle, when she has stopped spraying is approximately 3.08 m/s in the opposite direction.
And that's how we crunch the numbers in space, folks! Keep reaching for the stars (but maybe avoid the spray cans)!
To solve this problem, we can apply the law of conservation of momentum. The total momentum before spraying is equal to the total momentum after spraying.
Let's assign variables to the given quantities:
Mass of the astronaut (with spacesuit) = m₁ = 110 kg
Mass of the full spray can = m₂ = 20.8 kg
Mass of the spray released = m₃ = 5.0 kg
Speed of the spray released = v₃ = 16.4 m/s
Speed of the astronaut after spraying = v
Before spraying, the total momentum is zero since the astronaut and the can are at rest relative to the space shuttle.
Total momentum before spraying = m₁v₁ + m₂v₂
After spraying, the astronaut's speed will change, and the spray can will also have a speed relative to the astronaut.
Total momentum after spraying = (m₁ + m₂ + m₃)v
Now, using the conservation of momentum:
m₁v₁ + m₂v₂ = (m₁ + m₂ + m₃)v
Since the astronaut, relative to the space shuttle, is at rest (v₁ = 0), we can simplify the equation:
m₂v₂ = (m₁ + m₂ + m₃)v
Plugging in the given values:
(20.8 kg)(0 m/s) = (110 kg + 20.8 kg + 5.0 kg)v
Simplifying:
0 = (135.8 kg)v
Solving for v:
v = 0 m/s
The astronaut's speed, relative to the space shuttle, when she has stopped spraying, is approximately 0 m/s.
To solve this problem, we will use the principle of conservation of momentum. Momentum is defined as the product of an object's mass and its velocity.
Let's break down the problem step by step:
Step 1: Calculate the initial momentum of the astronaut and the can before any spraying occurs.
The initial momentum (p_initial) of the astronaut and the can combined can be calculated by adding their individual momenta. Since the astronaut and the can are initially at rest relative to the space shuttle, their initial velocities are zero.
p_initial = (mass_astronaut + mass_can) * 0
The mass of the astronaut is 110 kg, and the mass of the can is 20.8 kg.
p_initial = (110 kg + 20.8 kg) * 0 = 0
Step 2: Calculate the final momentum of the astronaut and the can after 5.0 kg of spray is released.
The mass of the remaining system after the spray is released is the sum of the mass of the astronaut and the remaining mass of the can, i.e., (mass_astronaut + mass_can - mass_spray). The velocity of the system after spraying is the velocity of the astronaut, which we need to find.
Let's denote the final velocity of the astronaut as v_final.
p_final = (mass_astronaut + mass_can - mass_spray) * v_final
The mass of the spray released is 5.0 kg. The remaining mass of the can is (mass_can - mass_spray), which is (20.8 kg - 5.0 kg).
p_final = (110 kg + (20.8 kg - 5.0 kg)) * v_final
= (126.8 kg) * v_final
Step 3: Apply the conservation of momentum.
According to the principle of conservation of momentum, the initial momentum and the final momentum of an isolated system remain the same.
Since the initial momentum was zero (0), the final momentum must also be zero (0).
0 = p_final = (126.8 kg) * v_final
Solving for v_final:
v_final = 0 / 126.8 kg
= 0 m/s
Therefore, the speed of the astronaut relative to the space shuttle when she stops spraying is 0 m/s.
her total mass = 110 + 20.8 - 5 = m
m v = 5 * 16.4