Please help with this calculus derivative problem?

Question

If a rock is thrown upward on the planet Mars with a velocity of 10 m/s, its height (in meters) after t seconds is given by H=10t-1.86t^2.
a) find the velocity of the rock after one second
b) find the velocity of the rock when t=a
c) when will the rock hit the surface?
d) with what velocity will the rock hit the surface?

(please explain your answer stepbystep so that I can learn it for my quiz today. i will give best answer to first correct response.. thanks!)

Answer 1

velocity is the derivative of position(height).

So, v=H'=10-(2)(1.86)t

Simply plug in t=1 and t=a for parts (a) and (b)

The rock will hit the surface when H=0
This occurs at two times. One is right when you throw it and is not what we are looking for. Solve H=0 for the other time.
H=10t-1.86t^2=0
divide both sides by t
10-1.86t=0

So... t=10/1.86 seconds
That covers part (c)

For part (d), use that time you find in part (c) and plug it into your equation for velocity just as you did in (a) and (b)

This gives you the velocity at time t=10/1.86 which is precisely the time that the rock hits the ground

Answer 2

a)The velocity of the rock is given by first derivative
v=dH/dt= 5-2*1.86*t= 5-3.72t +C for t=0 the constant is10
so v=15-3.72=12.28m/s

b) you replace t by a so v= 15-3.72a

c) The rock will rech its apex when v=0
so 0= 15-3.72 a a= 15/3.72=4.032s.
The height is H= 10*4.032*10-1.86*(4.032^2)=10.07m
to fall you must calculate the time it takes to fall of10.07m.the acceleration of gravity on Mars is 1.86m/s^2
so 10.07=1.86t^2 t^2=10/1.86=2.91s. so total time
T=4.032+2.91 =6.95s

d) velocity v = 1.86*2*t =1.86*2*2.91=
10.86m/s

Answer 3

a) v = dH/dt = 10 - 1.86 x 2t = 10 - 1.86 x2x1 = 10 - 3.72 = 6.28 m/s

b) v = dH/dt = 10 - 1.86 x 2t = 10 - 1.86 x2xa = (10 - 3.72a) m/s

c) Maximum height is when v = 0
v = dH/dt = 10 - 1.86 x 2t
0 = 10 - 3.72t hence t = 10/3.72
Total time to go up and come down = 2t = 2 x 10/3.72 = 20/3.72 = 5.376sec

d) v when hitting the surface remains same as when it was thrown upwards i.e 10 m/s