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The equation for a projectile's height versus time is h(t)=-16t^2+Vt+h. A tennis ball machine serves a ball vertically into the air from a height of 2 feet, with an initial speed of 130 feet per second. Which equation correctly models the ball's height as a function of time?

Respuesta :

carlosego
For this case we must use the projectile equation for this problem.
 We have then:
 h (t) = - 16t ^ 2 + Vt + h
 Where,
 V: initial speed
 h: height
 For the tennis ball we have:
 h (t) = - 16t ^ 2 + 130t + 2
 Answer:
 An equation that correctly models the ball's height as a function of time is:
 h (t) = - 16t ^ 2 + 130t + 2

The maximum height at the moment is 266.0625 ft.

Projectile's height

The equation for a projectile's height versus time is

[tex]&h(t)=-16 t^{2}+V t+h \\[/tex]

h = 2

V = 130

Substitute these values into the function

[tex]$h(t)=-16 t^{2}+130 t+2$[/tex]

Take the first derivative

[tex]$h^{\prime}(t)=-16* 2 t+130[/tex]

[tex]=-32 t+130$[/tex]

Equate the derivative with zero [tex]$h^{\prime}(t)=0$[/tex].

Solve the equation -32 t + 130=0

32 t = -130

t = -130 /(-32)

= 4.0625(s) .

Find the maximum height at the moment

t = 4.0625 (s)

h(4.0625) [tex]= -16(4.0625)^{2}+130 * 4.0625+2[/tex]

= - 264.0625+528.125+2

= 266.0625 ft.

Therefore, the maximum height at the moment is 266.0625 ft.

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