An astronaut of mass 60.0 kg and a small asteroid of mass 40.0 kg are initially at rest with res

An astronaut of mass 60.0 kg and a small asteroid of mass...

An astronaut of mass $60.0 \mathrm{kg}$ and a small asteroid of mass $40.0 \mathrm{kg}$ are initially at rest with respect to the space station. The astronaut pushes the asteroid with a constant force of magnitude $250 \mathrm{N}$ for $0.35 \mathrm{s}$. Gravitational forces are negligible. (a) How far apart are the astronaut and the asteroid $5.00 \mathrm{s}$ after the astronaut stops pushing? (b) What is their relative speed at this time?

Key Concepts

Newton's Second Law

This law relates the net force acting on an object to its mass and the acceleration that results from that force (F = ma). In problems like this, it allows us to calculate the acceleration of both the astronaut and the asteroid during the period when the force is applied.

Impulse-Momentum Theorem

The impulse-momentum theorem connects the impulse, which is the product of the force and the duration of its application, to the change in momentum of an object. This theorem is key for determining the velocity gained by both bodies as a result of the push.

Kinematics for Constant Acceleration

Kinematics equations for motion with constant acceleration are used to determine displacement and velocity over a period of time. After the force ceases, the bodies move at constant velocities, and these equations help calculate the distances each travels, allowing us to establish the separation and relative speed between them.

Relative Motion

Relative motion refers to the analysis of how objects move with respect to one another. In this problem, it is important for determining the separation distance and the relative speed between the astronaut and the asteroid after the force has been applied, by comparing their individual motions.

Transcript

00:02 For 171, there's an astronaut, and we know his mass, and there's an asteroid.

00:06 We knew that mass.

00:07 And the asteroid pushes off the asteroid with the force of 250 newtons, and they're in contact for 3 .5 seconds.

00:17 So, and then we're supposed to find how far apart are they, five seconds after being pushed.

00:24 So i can find the acceleration of both of them, just using newton cycle law, if equals ma, i know this forces on both of them, 250.

00:44 Astronaut pushes on asteroid.

00:48 Asteroid pushes on astronaut.

00:50 Newton's third loss is the same force on both of them.

00:54 So this 250 on the 60 kilogram astronaut, i can find that acceleration over here, the 250 on the 40 kilogram asteroid.

01:17 The acceleration of the astronaut is 4 .17 meters per second square.

01:30 And the acceleration of the asteroid is 6 .25.

01:38 It has less mass, so it's going to have more acceleration.

01:41 If that makes sense.

01:44 Okay.

01:46 So now i want to find how far they are after five seconds.

01:51 So after you're done pushing, then they're going to go at constant speed.

01:57 So if i can find that final speed that they're going after this 0 .35 seconds of pushing, then i can figure out how far they go.

02:06 So i'm going to make my list.

02:10 I know they're starting from rest, so v initial be zero.

02:14 I want to know what speed they've reached, because that's going to be the constant speed then.

02:21 I know the acceleration, so 4 .17.

02:26 And i know the time 0 .35 seconds.

02:41 So i'm gonna use the, i call it the a equation.

02:57 I usually do it in this form.

03:00 You might already have the a times the t multiplied and the v initial take into the other side.

03:09 So i'm gonna play main values and i get the final velocity here for the astronaut is 1 .46 meters per second.

03:40 Over here i do the same thing except this time over here my acceleration is this the 6 .25 i do all that same work and i get my v final for the asteroid is 2 .19.

04:07 Okay so now they're going constant speed after they're done being pushed so i can get just the constant speed equation v equals d or t.

04:22 Again, i'll show work for astronaut, and you know you do the same thing for asteroid.

04:28 So the speed it reached 1 .46.

04:33 I want to know how far it goes in five seconds now...

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