Finding a Constant Using Integration

[nivek516]

Someone is driving 350 ft/sec when he begins to put his foot on the brake to come to a complete stop. It takes a total of 1200 feet for him to stop. Find the constant of proportion, \(\displaystyle k\), for the acceleration given \(\displaystyle a=kt^2\), where \(\displaystyle a\) is acceleration and \(\displaystyle t\) is time in seconds.

I have started by integrating the equation to figure velocity \(\displaystyle (v=\frac{1}{3}kt^3+c)\), but I don't know how to figure out what \(\displaystyle c\) is. After I find \(\displaystyle c\), I can integrate the velocity equation to get displacement. Then, I am stuck on what to do.

 

[Deleted member 4993]

Someone is driving 350 ft/sec when he begins to put his foot on the brake to come to a complete stop. It takes a total of 1200 feet for him to stop. Find the constant of proportion, \(\displaystyle k\), for the acceleration given \(\displaystyle a=kt^2\), where \(\displaystyle a\) is acceleration and \(\displaystyle t\) is time in seconds.

I have started by integrating the equation to figure velocity \(\displaystyle (v=\frac{1}{3}kt^3+c)\), but I don't know how to figure out what \(\displaystyle c\) is. After I find \(\displaystyle c\), I can integrate the velocity equation to get displacement. Then, I am stuck on what to do.

Let us assume the car stops at t = t[subftge03h]f[/subftge03h]

so at t = t[subftge03h]f[/subftge03h], we have v=0 and s = 1200

and, when t = 0, v = 350 and s = 0

so c = 350

when t = t[subftge03h]f[/subftge03h], v = 0

k = -1050/t[subftge03h]f[/subftge03h][supftge03h]3[/supftge03h].................................(1)

now integrating v

S = k/12 * t[supftge03h]4[/supftge03h] + 350t + D

When t = 0, S = 0

so D = 0

Also when t = t[subftge03h]f[/subftge03h], S = 1200

1200 = k/12 * t[subftge03h]f[/subftge03h][supftge03h]4[/supftge03h] + 350t[subftge03h]f[/subftge03h].................(2)

Use (1) to solve for t[subftge03h]f[/subftge03h] - then solve for 'k'.

 

[nivek516]

Thank you so much! I posted different numbers just in case someone replied with the actual answer. I hate it when people do that...I like to learn it with description on how to solve. So, when the car travels 324 ft/s and takes 1093.5 ft to stop, I get a k value of -32/3. Correct? Thanks again!