Problem 1

A car starts from rest at time t = 0 and accelerates at -0.6t + 4 meters/sec^{2}for 0<t<12. How long does it take for the car to go 100 meters?

here is my work:

$\displaystyle a = -0.6t + 4 $

$\displaystyle v = -0.3t^2 + 4t + c $

$\displaystyle v = -0.3t^2 + 4t $

$\displaystyle s = -0.1t^3 + 2t^2 + K $

$\displaystyle s = -0.1t^3 + 2t^2 $

$\displaystyle 100 = -0.1t^3 + 2t^2 $

$\displaystyle t = 10 seconds $

Problem 2

A 727 jet needs to attain a speed of 200 mph to take off. If it can accelerate from 0 to 200 mph in 30 seconds, how long must the runway be? (Assume constant acceleration and give your final answer in feet)

here is my work:

200 miles per hour = $\displaystyle 293 \frac{1}{3} $ ft/sec

$\displaystyle 293 \frac{1}{3} $ ft/sec divided by 30 sec = $\displaystyle 9 \frac{7}{9} $ ft/sec$\displaystyle ^2$

$\displaystyle a = 9 \frac{7}{9} $ ft/sec$\displaystyle ^2$

$\displaystyle v = 9 \frac{7}{9} t + c $ ft/sec

$\displaystyle v = 9 \frac{7}{9} t $ ft/sec

$\displaystyle s = 4 \frac{8}{9} t^2 + K $

$\displaystyle s = 4 \frac{8}{9} t^2 $

$\displaystyle s(30) = 4 \frac{8}{9} * (30)^2 $

$\displaystyle s(30) = 4400 $ ft.

Please check the conversions and overall answer.

Problem 3

On the moon, the acceleration due to gravity is about 1.6 m/sec$\displaystyle ^2$ (compared to $\displaystyle g \approx 9.8 $ m/sec $\displaystyle ^2$ on earth). If you drop a rock on the moon (with initial velocity 0), find formulas for:

(a) Its velocity, v(t), at time t.

(b) The distance, s(t), it falls in time t.

here is my work:

$\displaystyle a = -1.6 $ m/sec$\displaystyle ^2$

$\displaystyle v = -1.6t + c $ m/sec

(a) $\displaystyle v(t) = -1.6t $ m/sec

(b) $\displaystyle s(t) = -0.8t^2 + K $, where K is the initial height.

Should I keep the constant K in this answer? and how about the constants in all the other problems? I'm confused whether I need them or not.. In the previous ones I assumed the initial values were 0. Can I do this?