Question

Can you please answer these questions but also add comments of your thought process in how you went about manipulating the equations?

A small rocket is escaping the Earth’s low orbit and accelerates from vi =8 km/s to velocity vf in a straight line traveling d=400 km while doing so. The acceleration achieved by its propulsion system is a constant 100 m/s2 during that time.

1. Using the definition of average speed in terms of vi and vf, write down the equation relating the distance travelled d to the time elapsed Dt and the final speed vf and vi

2. Using the definition of acceleration, write down an equation relating the acceleration to the elapsed time Dt, the final speed vf, and the initial speed vi.

3. Solve these equations for the final speed of the rocked after it has covered distance d=400 km.

Answer 1

given
vi = 8 km/s = 8000 m/s
a = 100 m/s^2
d = 400 km = 400000 m

as the rockect moves with constant acceleration we can use equations of motion.

i)

distance travelled = average speed*time taken

d = ((vi + vf)/2)*Dt

ii) acceleration = change in velocity/time taken

a = (vf - vi)/Dt

iii) multiply the above two equations

d*a = ((vi + vf)/2)*Dt*(vf - vi)/Dt

2*a*d = (vf + vi)*(vf - vi)

2*a*d = vf^2 -vi^2

==> vf^2 = vi^2 + 2*a*d

vf = sqrt(vi^2 + 2*a*d)

= sqrt(8000^2 + 2*100*400000)

= 12000 m/s

= 12 km/s <<<<<<<--------------------Answer