Question

A canister is dropped from a helicopter 500m above the ground. Its parachute does not open, but the canister has been designed to withstand an impact velocity of 100 m/s. Will it burst?

Answer 1

The impact speed 98.995 m/s is less than 100 m/s and the canister will not burst.

Explanation:

A function F is called an antiderivative of f on an interval I if
`F'(x) = f(x)` for all x in I.

Recall that if the object has position function
`s=f(t)`, then the velocity function is
`v(t)=s'(t)`. This means that the position function is an antiderivative of the velocity function. Likewise, the acceleration function is
`a(t)=v'(t)`, so the velocity function is an antiderivative of the acceleration.

An object near the surface of the earth is subject to a gravitational force that produces a downward acceleration denoted by
`g`. For motion close to the ground we may assume that
`g` is constant, its value being about
`9.8 \:{(m)/(s^2)}`.

We know that the acceleration due to gravity is given by

`a(t)=-9.8`

and the antiderivative is velocity

`v(t)=\int a(t)\,dt\\v(t)=\int -9.8\,dt\\v(t)=-9.8t +C`

We know that the canister was dropped, so the initial velocity at t = 0 is zero, this fact let us know the value of C.

`v(0)=9.8(0)+C\\C=0`

The antiderivative of velocity is the position

`s(t)=\int v(t) \, dt\\s(t)=\int -9.8t \, dt\\s(t)=-4.9t^2+C`

To find the value of the constant C, we know that the height was 500 m at t = 0, this means
`s(0)=500`

`500=-4.9(0)^2+C\\C=500`

`s(t)=-4.9t^2+500`

Using the fact that at the time of impact the height s(t) is zero we can compute the total time of the fall:

`s(t)=-4.9t^2+500=0\\\\-4.9t^2=-500\\\\t^2=(5000)/(49)\\\\\mathrm{For\:}x^2=f\left(a\right)\mathrm{\:the\:solutions\:are\:}x=√(f\left(a\right)),\:\:-√(f\left(a\right))\\\\t=\sqrt{(5000)/(49)},\:t=-\sqrt{(5000)/(49)}`

A negative time does not make sense, so we only take as a possible solution

`t=\sqrt{(5000)/(49)}=(50√(2))/(7)\approx 10.102`

Now the final velocity is

`v((50√(2))/(7))=-9.8((50√(2))/(7))\approx -98.995`

The impact speed 98.995 m/s is less than 100 m/s and the canister will not burst.