91Ó°ÊÓ

Acceleration is a fundamental concept in kinematics that describes the rate of change of velocity over time. In simpler terms, it's how quickly an object speeds up or slows down. For the spacecraft in the problem, the engines firing cause a change in velocity, thereby leading to acceleration. In this exercise, we find that the spacecraft has two acceleration components: one along the x-axis and one along the y-axis. These components tell us how the speed in each direction changes due to the engines. For example, a higher acceleration value in the x direction means the spacecraft is speeding up more quickly horizontally.

• The formula for finding acceleration components is derived from the kinematic equation.
• The accelerations are calculated by comparing initial velocity, displacement, and time.

Velocity components refer to the parts of an object's velocity that occur in specific directions. In the context of kinematics, it's often helpful to break down velocity into horizontal (x-component) and vertical (y-component) parts.The initial velocity components given in the exercise are:

• Horizontal velocity, \( v_{0x} = 4370 \, \text{m/s}\)
• Vertical velocity, \( v_{0y} = 6280 \, \text{m/s}\)

These values indicate how fast the spacecraft was moving horizontally and vertically before the engines were fired. By considering the separate components, it becomes easier to manage and solve for how each direction changes independently during the motion. This separation into components is crucial when analyzing movements in different dimensions.

Displacement is the vector quantity that describes the change in position of an object. It contains both the magnitude and the direction of the movement.In the given problem, the spacecraft's displacement components are provided as:

• Horizontal displacement, \( x = 4.11 \times 10^6 \, \text{m} \)
• Vertical displacement, \( y = 6.07 \times 10^6 \, \text{m} \)

These values show how far the spacecraft has moved along the x and y axes while the engines were engaged. The displacement helps to calculate the effect of the firing engines over time, showing how much ground is covered in both the horizontal and vertical axes. Understanding displacement is key to analyzing motion, as it directly relates to finding out how velocity changes.

Kinematic equations are mathematical formulas used to describe the motion of objects under constant acceleration. They relate displacement, velocity, acceleration, and time, making them invaluable for solving problems like the one in the exercise.The specific equation used here is:\[ s = v_0 t + \frac{1}{2} a t^2 \]This equation helps calculate an object's displacement based on its initial velocity, acceleration, and the time period over which these occur.

• The equation is applied separately to each direction: x and y.
• It enables solving for unknown acceleration by rearranging the formula.

By leveraging the kinematic equations, one can determine acceleration values when certain components like initial velocity, displacement, and time are known. This approach simplifies the understanding and calculation of complex motions.