91Ó°ÊÓ

Displacement in kinematics refers to the change in position of an object. It is a vector quantity, which means it has both magnitude and direction. In the context of the spacecraft problem, displacement components in both the x and y directions are given:

• An x-component of \(4.11 \times 10^6 \, \text{m} \)
• A y-component of \(6.07 \times 10^6 \, \text{m} \)

Displacement differs from distance, as displacement only considers the starting and ending positions, while distance would consider the total path traveled. For this problem, knowing displacement helps in calculating the acceleration of the spacecraft once the initial velocity and time are known.

Velocity is also a vector quantity and can be broken down into components, typically in the x and y directions in two-dimensional motion. These components are essential as they allow for separate analysis of motion in each direction.

• The initial x-component of velocity for the spacecraft is \(v_{0x} = 4370 \, \text{m/s} \).
• The initial y-component of velocity is \(v_{0y} = 6280 \, \text{m/s} \).

By understanding velocity components, you can solve complex motion problems by analyzing each direction separately. These components help determine the effects of external forces, like the engines in this problem, on the spacecraft's motion over the given time period of 684 seconds.

Acceleration, like velocity and displacement, is a vector and can also be expressed in components. It describes the rate of change of velocity with time. In the given exercise, the task is to find the x and y components of acceleration.To find these:- Use the kinematic equations that relate displacement, initial velocity, and acceleration over time.- The computations, as seen in the solution, involve rearranging the displacement formula to solve for the respective acceleration components.Knowing the acceleration components (\(a_x \approx 4.78 \, \text{m/s}^2 \) and \(a_y \approx 7.58 \, \text{m/s}^2 \)), gives us insight into how the spacecraft's velocity changes over time due to its engines.

Kinematic equations are essential mathematical tools used to describe the motion of objects. They relate the five key motion variables: displacement (s), initial velocity (\(v_0\)), final velocity (v), acceleration (a), and time (t).For scenarios like the spacecraft problem, the key kinematic equation used is:\[s = v_0 t + \frac{1}{2} a t^2\]This equation links initial velocity, acceleration, and displacement over a specific time, allowing us to solve for unknown variables when the other quantities are known.Apply these equations:- For the x-direction: \[4.11 \times 10^6 = 4370 \times 684 + \frac{1}{2} a_x \times 684^2\]- For the y-direction: \[6.07 \times 10^6 = 6280 \times 684 + \frac{1}{2} a_y \times 684^2\]These steps in the solution demonstrate how to isolate and solve for acceleration, illustrating the powerful application of kinematic equations in physics.