Respuesta :
Answer:
The impact speed 98.995 m/s is less than 100 m/s and the canister will not burst.
Step-by-step explanation:
A function F is called an antiderivative of f on an interval I if [tex]F'(x) = f(x)[/tex] for all x in I.
Recall that if the object has position function [tex]s=f(t)[/tex], then the velocity function is [tex]v(t)=s'(t)[/tex]. This means that the position function is an antiderivative of the velocity function. Likewise, the acceleration function is [tex]a(t)=v'(t)[/tex], 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 [tex]g[/tex]. For motion close to the ground we may assume that [tex]g[/tex] is constant, its value being about [tex]9.8 \:{\frac{m}{s^2}}[/tex].
We know that the acceleration due to gravity is given by
[tex]a(t)=-9.8[/tex]
and the antiderivative is velocity
[tex]v(t)=\int a(t)\,dt\\v(t)=\int -9.8\,dt\\v(t)=-9.8t +C[/tex]
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.
[tex]v(0)=9.8(0)+C\\C=0[/tex]
The antiderivative of velocity is the position
[tex]s(t)=\int v(t) \, dt\\s(t)=\int -9.8t \, dt\\s(t)=-4.9t^2+C[/tex]
To find the value of the constant C, we know that the height was 500 m at t = 0, this means [tex]s(0)=500[/tex]
[tex]500=-4.9(0)^2+C\\C=500[/tex]
[tex]s(t)=-4.9t^2+500[/tex]
Using the fact that at the time of impact the height s(t) is zero we can compute the total time of the fall:
[tex]s(t)=-4.9t^2+500=0\\\\-4.9t^2=-500\\\\t^2=\frac{5000}{49}\\\\\mathrm{For\:}x^2=f\left(a\right)\mathrm{\:the\:solutions\:are\:}x=\sqrt{f\left(a\right)},\:\:-\sqrt{f\left(a\right)}\\\\t=\sqrt{\frac{5000}{49}},\:t=-\sqrt{\frac{5000}{49}}[/tex]
A negative time does not make sense, so we only take as a possible solution
[tex]t=\sqrt{\frac{5000}{49}}=\frac{50\sqrt{2}}{7}\approx 10.102[/tex]
Now the final velocity is
[tex]v(\frac{50\sqrt{2}}{7})=-9.8(\frac{50\sqrt{2}}{7})\approx -98.995[/tex]
The impact speed 98.995 m/s is less than 100 m/s and the canister will not burst.
