Landing Rockets on Ships

Classical Mechanics Level 3

SpaceX is trying to land their next reusable rocket back on a drone ship. The drone ship is traveling due west in an ocean current at a constant speed of $5 \text{ m}/\text{s}$ . The rocket is $2000 \text{ m}$ east of the drone ship and $5000 \text{ m}$ vertically above it, traveling vertically downwards at $100 \text{ m}/\text{s}$ . If the rocket can apply vertical and horizontal thrusts to change the acceleration of the rocket in the vertical and horizontal directions, and must accelerate constantly in both directions, find the magnitude of the net acceleration (vector) required to land on the ship with no vertical velocity. Answer in $\text{ m}/\text{s}^2$ to the nearest tenth.

The answer is 1.1.

1 solution

Matt DeCross
May 1, 2016

The acceleration has two components, $a_x$ and $a_y$ . Equating the horizontal distances traveled by the rocket and ship in time $t$ yields:

$2000+5t = \frac12 a_x t^2.$

Since the ship ends at rest, the third equation of motion for the vertical acceleration is:

$0 = 100^2- 2a_y (5000).$

The time of motion can be found by the constraint that the ship ends at rest:

$a_y t = 100.$

From the equation for $a_y$ , one finds $a_y = 1$ . Using the ending at rest constraint then gives $t = 100$ . Finally, equating the horizontal distances gives the horizontal acceleration $a_x = \frac12$ .

From this we have the answer:

$|a| = \sqrt{1 + \left(\frac12\right)^2} \approx 1.1.$

Units have been omitted above but should be obvious in context.

Hi Matt,

With this acceleration, the vertical velocity is 0, but the horizontal one is 50 m/s, faster than the drone's. Elon wouldn't be happy about it.

Hari Kumar - 2 years, 1 month ago

Of course, it's a simple problem to test kinematics :) In reality you would want to lower the rocket slowly to the drone ship at the ship's horizontal speed, at the end, which would require a faster horizontal acceleration beforehand.

Matt DeCross - 1 year, 10 months ago

It was a joke Matt... They wouldn't do such an obvious one..

Hari Kumar - 1 year, 10 months ago

Landing Rockets on Ships

The answer is 1.1.

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