Motion of spaceship. A spaceship is floating in free space with an initial velocity \(\mathbf{v}_{0}=1000 \mathrm{~m} / \mathrm{s} \mathbf{i}\). Suddenly, the spaceship turns its thrusters on, giving the spaceship a constant acceleration \(\mathbf{a}_{0}=10 \mathrm{~m} / \mathrm{s}^{2} \mathbf{j}\) for \(10 \mathrm{~s}\). (The acceleration does not change during this time). (a) Sketch the path of the spaceship without doing a detailed calculation. (b) Find the velocity of the spaceship as a function of time. (c) Find the position of the spaceship as a function of time. (d) Plot the path of the spaceship and compare with your initial sketch.

In this problem, our spaceship experiences constant acceleration. Constant acceleration means the rate of change of velocity remains the same over time. The spaceship's thruster provides this constant force perpendicular to its initial motion. This changes the velocity steadily in that direction. The important takeaway here is that acceleration is a vector quantity, meaning it has both magnitude and direction. For our spaceship, the acceleration is given as \(\text{10 m/s}^2\) in the j-direction (or y-axis). This unchanging acceleration over the 10 seconds is key to understanding the resulting motion.

Now, let's dive into the velocity function. Velocity at any time is found by adding the initial velocity to the change in velocity caused by acceleration. We start with the initial velocity of \(\text{1000 m/s \mathbf{i}}\text{(in the x-direction)}\). We've got a constant acceleration of \(\text{10 m/s}^2\) in the y-direction (j).

Over time, this acceleration changes the y-component of the velocity. So the velocity function is:
\[\text{\textbf{v}(t) = \textbf{v}_{0} + \textbf{a} t}\]
Breaking this down:
\(\text{\textbf{v}(t) = 1000 \mathbf{i} + 10t \mathbf{j}}\text{ m/s}\).
On plugging in any time t into this function, we get the spaceship's velocity at that point in time. This helps imagine and calculate the changing speed and direction over the 10 seconds.

Position function tells us where the spaceship is at any given time based on its velocity. Since velocity is the rate of change of position, the position can be found by integrating the velocity function over time. Starting from an initial position of zero \(\text{\textbf{r}_{0}}\text{=0 at t=0}\), we integrate the velocity components:
\(\text{\textbf{r}(t) = \text{\textbf{r}}_{0} + \text{\textbf{v}}_{0} t + \frac{1}{2} \text{\text{\textbf{a}}}_{0} t^2}\).

For our spaceship:
\(\text{\textbf{r}(t)= 1000t \mathbf{i} + 5t^2 \mathbf{j}}\text{ m}\).
This function tells us the x and y coordinates of the spaceship at any time t. The x-component increases linearly (1000t), while the y-component grows quadratically (5t^2). This results in a parabolic path.

The entire problem here is an example of 2D motion analysis. We have motion in two perpendicular directions (x and y). In the x-direction, there's no acceleration, only the initial velocity \(\text{1000 m/s \mathbf{i}}\).

In the y-direction, the spaceship starts from rest and accelerates constantly at \(\text{10 m/s}^2\). These two motions combine to create a parabolic path.

When we plot this, we see the horizontal distance increasing at a steady rate, while the vertical distance starts slow and then increases more sharply over time. This visualization confirms the path being a parabola opening upwards starting from a purely horizontal initial motion.