Question

A spaceship is approaching Starbase Beta at an initial velocity of 130 km/s in the +x direction. The ensign sets its computer to initiate a braking program starting at time t = 0. After this time, the computer controls the ship until the ship docks at time t= T so that its x-acceleration is ax(t) = b(t-T), where b = 0.26 m/s3. Note that ax < 0 for t < T, but gets smaller as t approaches the docking time T.

a) What must T be so that the ship's x-velocity is also zero at the time of docking?

b) How far is the ship from the station at t=0?

Answer 1

The time T at which the ship's x-velocity is zero can be found by solving an equation. The ship must dock at t = 7.7 seconds. The ship is 0 km from the station at t = 0.

Step-by-step explanation:

To find the time T at which the ship's x-velocity is zero, we can set the final x-velocity to zero and solve for T. The final x-velocity is given by the integral of ax(t) over the interval [0, T]. The integral of b(t-T) with respect to t is b/2 * (t-T)^2. Setting this equal to zero and solving for T gives:

T = 2T = 0.26(T^2)

Simplifying, we get:

T^2 - 7.7T = 0

Factoring out T, we get:

T(T - 7.7) = 0

So, either T = 0 or T = 7.7. Since T cannot be zero, the ship must dock at t = 7.7 seconds.

To find the distance of the ship from the station at t = 0, we can use the formula for distance traveled under constant acceleration, which is given by d = v_i * t + 0.5 * a * t^2, where v_i is the initial velocity and a is the acceleration.

Since the ship is traveling at a constant velocity of 130 km/s, the distance traveled at time t = 0 is given by d = 130 km/s * 0 + 0.5 * 0 * 0^2 = 0.

So, the ship is 0 km from the station at t = 0.