Question

The height h in feet of a baseball on Earth after t seconds can be modeled by the function h(t) = -16(t - 1.5)2 + 36, where -16 is a constant in ft/s2 due to Earth's gravity. The gravity on Mars is only 0.38 times that on Earth. If the same baseball were thrown on Mars, it would reach a maximum height 59 feet higher and 2.5 seconds later than on Earth. Write a height function for the baseball thrown on Mars.

Answer 1

A short stop makes an error by dropping the ball. As the ball drops, its height h in feet as a function of time, t, is modeled by h(t) = -16t2 + 3. A slow motion replay of the error shows the play at half speed. What function describes the height of the ball in the replay? (Hint: The function squares the time, so half the time is also squared in the new function.)

Answer 2

h(t) = -16(t -1.5)² + 36
h(2.5) = -16(2.5-1.5)² + 36
h(2.5) = -16(2.5²-1.5²) + 36
h(2.5) = -16 + 36
h(2.5) = 20

The height on Earth is 20 ft at 2.5 seconds.

20+59=79
2.5+2.5=5

The height on Mars would be 79 ft. at 5 seconds.

Answer 3

Write a height function for the baseball thrown on Mars.

First, they tell you that on Mars the baseball reaches a max height at a certain time. This will be the \( vertex \) of a parabola. That is a clue to use the vertex form of the equation of a parabola.

h(t) = a(x -h) + k where (h,k) is the vertex

see http://www.mathwarehouse.com/geometry/parabola/standard-and-vertex-form.php

Answer 4

so you want to figure out the vertex on Mars. The max height is 59 feet higher than on Earth. What is the max height on Earth ?

Look at Earth's equation:
\[ h(t) = -16(t - 1.5)^2 + 36 \]
This is also in vertex form, so you know that vertex (for Earth) is (1.5, 36)
The max height on Earth is 36.

Mar's max height is 59 feet higher than on Earth, so 59+36= 95
this happens 2.5 seconds later than on Earth. On Earth the time is 1.5 seconds, so on Mars, it is 2.5+1.5= 4 seconds

the vertex on Mars is (4,95)

use this to write the equation in vertex form:
\[ h(t)= a(t-4)^2 + 95 \]

Almost done: you need to find "a"
on Earth, "a" is -16. (see Earth's vertex equation)

They tell you -16 is a constant in ft/s2 due to Earth's gravity
and
gravity on Mars is only 0.38 times that on Earth.
this means "a" for Mars is 0.38* -16 = -6 (roughly)

Answer 5

The final equation is
\[ h(t)= -6(t-4)^2 + 95 \]