Question

Our hero, Captain Gravity, is being pulled into a black hole at an acceleration of 20 mi / h2, where x is the distance between his ship and the black hole in miles. Right now he is 300 miles away, but he was not paying attention to his instruments, so he is already moving toward the black hole at 100 mi / h. His thrusters give him a constant acceleration of 30 mi / h2 in any direction. Evaluate the following as true or false. Captain Gravity will get sucked into the black hole.

Answer 1

True

Step-by-step explanation:

We need to know the net acceleration of Captain Gravity, since it will depend on it to be saved from being swallowed by the black hole:

`a_(net)=a_b+a_c\\a_(net)=20(mi)/(h^2)-30(mi)/(h^2)\\a_(net)=-10(mi)/(h^2)`

Where
`a_b` is the acceleration with which Captain Gravity is being pulled into the black hole and
`a_c` is the deceleration due to his thrusters.

The captain will be swallowed if the distance necessary for him to decelerate until he reaches a speed equal to 0 is greater than 300 miles, since this is his distance to the black hole. Then using the kinematic equations we can know the value of this distance d:

`v_f^2=v_i^2+2ad`

Rewriting for d:

`2ad=v_f^2-v_i^2\\d=(v_f^2-v_i^2)/(2a)\\d=(0-(100(mi)/(h))^2)/(2(-10(mi)/(h^2)))\\d=(-10000(mi)/(h))/(-20(mi)/(h^2))=500mi`

Since the required distance
`d` is greater than the distance
`x`, unfortunately our gravity captain will get sucked into the black hole.