People hoping to travel to other worlds are faced with huge...

Key Concepts

Uniformly Accelerated Motion

This concept relates to motion under a constant acceleration, where the velocity of an object increases linearly with time. In the context of space travel, calculating the distance covered during the acceleration phase involves using kinematic equations, such as s = 0.5 × a × t², which is essential when a spacecraft accelerates at a constant rate such as 1 g.

Constant Velocity Motion

Once the spacecraft stops accelerating, it travels at a constant speed. This phase of the journey is governed by the basic relationship between distance, speed, and time (s = v × t). Understanding constant velocity motion is key to calculating the remaining time needed to cover the vast distances between stars.

Time-Distance Relationships in Kinematics

This principle involves using the standard equations of motion to relate time, distance, acceleration, and velocity. In planning interstellar journeys, one must combine the periods of acceleration and constant speed travel to determine the total time taken for the trip.

Interstellar Distances

Interstellar distances refer to the enormous separations between stars, typically measured in meters, kilometers, or astronomical units. Recognizing the scale of these distances is crucial when evaluating the challenges of space travel, as even a journey to the nearest star requires dealing with extraordinarily lengthy distances.

Transcript

0:00 Okay.

00:02 So this question, we have a spaceship that's trying to travel to a star that's 4 .1 times 10 to 16 meters away from earth.

00:10 And we're going to be accelerating at 1g for half a year.

00:13 So this kind of technology doesn't exist, but we're going to do it for the problem's sake.

00:20 And another thing that's useful that i looked up is that half a year is 1 .5 times 10 to the 7 seconds.

00:27 So we're often working in units of meters per second.

00:31 I saw a unit, so it's going to be easier to work in seconds than years to help stay consistent.

00:37 And basically when we draw out our velocity versus time graph, what we see, we're going to see a constant acceleration for half a year.

00:50 And then we're going to reach a constant velocity and continue onwards until we reach that star.

00:56 So there's a couple different ways to approach the problem.

00:59 The way that i'm going to go about doing it is first, first, i'm going to calculate the distance traveled in this region.

01:12 So while we're accelerating, how far are we going to go? so i'm going to call this r1, r1.

01:21 So calculate r1.

01:25 Okay.

01:26 And r1 will be useful because then from r1 we can then find the distance traveled in this region once we're at constant velocity, because we know the total distance.

01:43 The total distance is going to be r1 plus r2.

01:49 R1 plus r2.

01:50 So we can find r2 really easily.

01:53 And from r2, we can calculate the time it takes to travel.

02:00 So we'll call this in the distance r2, we're going to travel time 2.

02:06 We're going to calculate time 2.

02:09 And because the question is asking us for the total amount of time, we are going to add time two to the time of half a year.

02:26 And that's going to give us a total journey time.

02:28 And lastly, the one thing that i forgot to mention is we're going to calculate this distance, but we're also going to need to know v final.

02:38 So this speed here, v final.

02:44 And that's going to allow us to then use our r2 distance to calculate t2.

02:52 So that's a value that we're going to need.

02:55 And so let's start off.

02:57 So the equation that i'm going to use is our displacement.

03:04 So we know that r1 is going to equal r0 plus our initial velocity times time plus one -half acceleration times times squared.

03:15 Okay, we know the acceleration, we know the time, we know that our initial velocity is zero, we know that our initial position starting from earth is zero so then we're going to get something like this we're going to have r1 equals one half 1g is 9 .8 meters per second squared times time.

03:37 Okay, 1 .57 times 10 to the 7 squared second squared and that's going to give us an r1 and of 1 .218 times 10 to the 15 meters.

03:54 Wow.

03:55 Okay.

03:57 So now we have our r1 and with r1 we can also calculate v final.

04:05 And v final is going to be that acceleration times time plus the initial.

04:14 So we actually don't need r1.

04:15 But the final is then going to be that acceleration times time plus the initial.

04:15 So we actually don't need r1.

04:16 But the final is then going to be that 9...