Initial Value

[TwistedNerve]

My prof said this was an initial value problem. There are hardly any examples in the textbook.

Here it is

A jet plane lands at 160 miles per hour on a runway. If the plane brakes/decelerates at a constant rate, find the minimal such rate required to land on a 7000 ft runway.


So I let x be the acceleration, and found the antiderivative and got .5x^2 + Vnaught (160 mph, or 234 ft/s). Then I took the antiderivate of that and got (1/6)x^3 + 234x + C where C would be the length of the airstrip, or 7000 feet. I got 22 ft/s/s as the answer (graphed and found zero) but that doesnt seem right. Is 22 the time it takes for the plane to stop? even that seems wrong (too small). what did I do wrong?

I know there are various physic equations but Im curious as to how its done using calculus. Thanks

 

[daon]

The acceleration is described as constant and should not be treated as a variable. So if \(\displaystyle a\) is the acceleration and you are descibing time with the variable \(\displaystyle x\), the antiderivative is \(\displaystyle ax+v_o\) for some constant \(\displaystyle v_o\).

 

[Deleted member 4993]

My prof said this was an initial value problem. There are hardly any examples in the textbook.

Here it is

A jet plane lands at 160 miles per hour on a runway. If the plane brakes/decelerates at a constant rate, find the minimal such rate required to land on a 7000 ft runway.


So I let x be the acceleration, and found the antiderivative and got .5x^2 + Vnaught (160 mph, or 234 ft/s). Then I took the antiderivate of that and got (1/6)x^3 + 234x + C where C would be the length of the airstrip, or 7000 feet. I got 22 ft/s/s as the answer (graphed and found zero) but that doesnt seem right. Is 22 the time it takes for the plane to stop? even that seems wrong (too small). what did I do wrong?

I know there are various physic equations but Im curious as to how its done using calculus. Thanks

Since this is a constant acceleration problem - Galilieo devised three equations to solve these types of problems:
\(\displaystyle v = u + at\)

\(\displaystyle s = s_0 + ut + 1/2 at^2\)

\(\displaystyle v^2 = u^2 + 2 as\)

where

u = initial speed

v = final speed

s = distance travelled

t = time expired

If you don't have to use calculus, use the third equation and find 'a'.