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marysya [2.9K]
3 years ago
13

A canister is dropped from a helicopter 500m above the ground. Its parachute does not open, but the canister has been designed t

o withstand an impact velocity of 100 m/s. Will it burst?
Mathematics
1 answer:
solmaris [256]3 years ago
5 0

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<em> F</em> is called an antiderivative of <em>f</em> on an interval <em>I</em> if F'(x) = f(x) for all x in <em>I.</em>

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 \:{\frac{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=\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}}

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

t=\sqrt{\frac{5000}{49}}=\frac{50\sqrt{2}}{7}\approx 10.102

Now the final velocity is

v(\frac{50\sqrt{2}}{7})=-9.8(\frac{50\sqrt{2}}{7})\approx -98.995

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

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