Final answer:
A rock is thrown upward on the planet Mars with a velocity of 15 m/s. After one second, the velocity of the rock is 11.28 m/s. The rock will hit the surface after approximately 8.1 seconds with a velocity of -1.98 m/s.
Step-by-step explanation:
(a) Find the velocity of the rock after one second.
To find the velocity of the rock after one second, we need to substitute t = 1 into the velocity equation. The velocity equation is given by:
v = dH/dt
where H is the height equation. By differentiating the equation H = 15t − 1.86t2 with respect to t, we get:
v = d(15t − 1.86t2)/dt = 15 - 3.72t
Substituting t = 1 into the equation, we get:
v = 15 - 3.72(1) = 11.28 m/s
Therefore, the velocity of the rock after one second is 11.28 m/s.
(b) Find the velocity of the rock when t = a.
To find the velocity of the rock when t = a, we substitute t = a into the velocity equation. So, the velocity of the rock when t = a is given by:
v = 15 - 3.72a
(c) When will the rock hit the surface?
The rock will hit the surface when the height is equal to 0. So, we set H = 0 and solve for t:
15t - 1.86t2 = 0
This is a quadratic equation that can be solved by factoring or using the quadratic formula. Solving for t, we get:
t = 0 or t = 8.06
Since time cannot be negative, the rock will hit the surface after approximately 8.1 seconds.
(d) With what velocity will the rock hit the surface?
To find the velocity at which the rock will hit the surface, we substitute t = 8.06 into the velocity equation:
v = 15 - 3.72(8.06) = -1.98 m/s
Therefore, the rock will hit the surface with a velocity of approximately -1.98 m/s.