A spaceship approaches Starbase Beta with an initial velocity moving in the +x direction. The ensign sets the ship’s computer to initiate a braking program starting at time t = 0. After this time the computer controls the ship’s x-velocity according to until the ship docks at the Starbase. The numerical values of the constants are b = 100 and (alpha) = 0.1 (SI units).


(a)Sketch vx(t). What are the units of b and (alpha)? Interpret what they represent physically.

(b)Find ax(t) and sketch it. When is the maximum acceleration and what is its value (magnitude)?

(c)Find x(t) and sketch it. Assume that the position of the spaceship is defined to be at x = 0 at t = 0. How far has the ship traveled after it has come to rest?

To find the answers to the questions, we will need to use equations of motion and calculus.

(a) The equation to describe the velocity of the spaceship at any time t is given by:

v_x(t) = v_i - bt

Where:
v_x(t) is the velocity of the spaceship at time t
v_i is the initial velocity of the spaceship (in the +x direction)
b is a constant with units m/s^2, which represents the rate of deceleration
t is the time

From the given information, the spaceship has an initial velocity in the +x direction, so v_i is positive. The constant b represents the rate at which the ship is slowing down. The unit for b is m/s^2, which represents how many meters per second the ship's velocity decreases every second. Therefore, b represents the deceleration of the spaceship.

To sketch v_x(t), plot the velocity as a function of time on a graph. Start with the initial velocity v_i at t = 0, and then draw a straight line with a negative slope (representing the deceleration) until the velocity reaches zero when the spaceship docks at the Starbase.

(b) The acceleration of the spaceship can be found by taking the derivative of the velocity function with respect to time:

a_x(t) = -b

The acceleration, a_x(t), is a constant value of -b, which means it is a negative value representing deceleration. The maximum acceleration occurs at the beginning when the ship starts braking, and its value is -b (in magnitude).

To sketch a_x(t), plot the acceleration as a function of time on a separate graph. It will be a horizontal line with a constant value of -b.

(c) The position of the spaceship can be found by integrating the velocity function with respect to time:

x(t) = ∫[v_i - bt] dt

Integrating v_x(t), we get:

x(t) = v_i * t - (1/2) * b * t^2

Given that the position of the spaceship is defined to be at x = 0 at t = 0, we can use this information to find an expression for x(t).

To sketch x(t), plot the position as a function of time on another graph. It will initially increase with a positive quadratic curve until it reaches a maximum value and starts decreasing eventually reaching zero again when the spaceship docks at the Starbase.

To find how far the ship has traveled after it has come to rest, we need to find the time when the spaceship comes to rest. This occurs when v_x(t) = 0. Solving the equation v_i - bt = 0 for t, we find t = v_i / b.

Substituting this time into the position function x(t), we can find the distance traveled:

x_final = v_i * t - (1/2) * b * t^2

Where x_final represents how far the spaceship has traveled.