A 130 kg astronaut (including space suit) acquires a speed of 2.80 m/s by pushing off with his legs from an 2000 kg space capsule.
What is the change in speed of the space capsule? As the reference frame, use the position of the space capsule before the push.
If the push lasts 0.43 s, what is the average force exerted on the astronaut by the space capsule?
To find the change in speed of the space capsule, we can apply the principle of conservation of momentum. According to this principle, the total momentum of an isolated system remains constant if no external forces act on it.
Before the push, the momentum of the system (astronaut + space capsule) is zero since both are at rest. After the push, the momentum of the system must also be zero since there are no external forces acting on it.
Now let's calculate the initial and final momenta separately:
1. Initial momentum of the system:
- The initial momentum of the astronaut is given by:
momentum_astro_initial = mass_astro * velocity_astro_initial
- The initial momentum of the space capsule is:
momentum_capsule_initial = mass_capsule * velocity_capsule_initial
Since both the astronaut and the space capsule are at rest initially, their initial velocities are zero. Therefore, the initial momenta of both the astronaut and the space capsule are zero.
2. Final momentum of the system:
- The final momentum of the astronaut is given by:
momentum_astro_final = mass_astro * velocity_astro_final
- The final momentum of the space capsule is:
momentum_capsule_final = mass_capsule * velocity_capsule_final
Since the total momentum of the system (astronaut + space capsule) is zero before and after the push, we can write the equation:
momentum_astro_initial + momentum_capsule_initial = momentum_astro_final + momentum_capsule_final
Since the initial momenta of both the astronaut and the space capsule are zero, the equation simplifies to:
momentum_astro_final + momentum_capsule_final = 0
Now let's substitute the given values and solve for the final velocity of the space capsule:
momentum_astro_final + momentum_capsule_final = 0
mass_astro * velocity_astro_final + mass_capsule * velocity_capsule_final = 0
Substituting the given values:
130 kg * velocity_astro_final + 2000 kg * velocity_capsule_final = 0
We know that the velocity of the astronaut after the push is 2.80 m/s, so we can substitute that value:
130 kg * 2.80 m/s + 2000 kg * velocity_capsule_final = 0
Simplifying the equation:
364 kg·m/s + 2000 kg · velocity_capsule_final = 0
Now, solving for velocity_capsule_final:
velocity_capsule_final = - ( 364 kg·m/s ) / 2000 kg
Therefore, the change in speed of the space capsule is equal to the final velocity of the space capsule, which is approximately -0.182 m/s. The negative sign indicates that the space capsule's speed has decreased.
To find the average force exerted on the astronaut by the space capsule during the push, we can use Newton's second law of motion, which states that the force acting on an object is equal to the rate of change of its momentum.
The average force exerted on the astronaut is given by:
average_force = change_in_momentum / time
The change in momentum of the astronaut can be calculated as:
change_in_momentum = momentum_astro_final - momentum_astro_initial
Since the initial momentum of the astronaut is zero, the equation simplifies to:
change_in_momentum = momentum_astro_final
Substituting the given values, we can find the change in momentum:
change_in_momentum = mass_astro * (velocity_astro_final - velocity_astro_initial)
= 130 kg * (2.80 m/s - 0 m/s)
= 130 kg * 2.80 m/s
= 364 kg·m/s
Now, substituting the change in momentum and the time into the equation for average force:
average_force = change_in_momentum / time
= (364 kg·m/s) / 0.43 s
= 846.5 N
Therefore, the average force exerted on the astronaut by the space capsule during the push is approximately 846.5 Newtons.