Question

A projectile is launched straight up from a height of 960 feet with an initial velocity of 64 ft/sec. Its height at time t is h(t)=−16t2+64t+960. Answer the following questions, including units in each answer. At what time does it reach its maximum height? Hint: Its velocity is zero at its maximum height. How high will it go? What is its velocity at the instant it reaches the ground?

Answer 1

a)
`t=2s`

b)
`h_(max)=1024ft`

c)
`v_(y)=-256ft/s`

Step-by-step explanation:

From the exercise we know the initial velocity of the projectile and its initial height

`v_(y)=64ft/s\\h_(o)=960ft\\g=-32ft/s^2`

To find what time does it take to reach maximum height we need to find how high will it go

b) We can calculate its initial height using the following formula

Knowing that its velocity is zero at its maximum height

`v_(y)^(2)=v_(o)^(2)+2g(y-y_(o))`

`0=(64ft/s)^2-2(32ft/s^2)(y-960ft)`

`y=(-(64ft/s)^2-2(32ft/s^2)(960ft))/(-2(32ft/s^2))=1024ft`

So, the projectile goes 1024 ft high

a) From the equation of height we calculate how long does it take to reach maximum point

`h=-16t^2+64t+960`

`1024=-16t^2+64t+960`

`0=-16t^2+64t-64`

Solving the quadratic equation

`t=\frac{-b±\sqrt{b^(2)-4ac}}{2a}`

`a=-16\\b=64\\c=-64`

`t=2s`

So, the projectile reach maximum point at t=2s

c) We can calculate the final velocity by using the following formula:

`v_(y)^(2)=v_(o)^(2)+2g(y-y_(o))`

`v_(y)=±\sqrt{(64ft/s)^(2)-2(32ft/s^2)(-960ft)}=±256ft/s`

Since the projectile is going down the velocity at the instant it reaches the ground is:

`v=-256ft/s`