Question

Bob is 70 miles away from home. He starts his trip home going 70 mph, and slows down continuously.
(a) He plans to stop at the store one mile away from home. How long will it take him to get to the store?
(b) How long will it take him to get from the store to his home if he uses the same driving strategy?

Answer 1

V=70-at

I assume at he slows down continuously and stopping at his home.
Let's use a kinematic equation
vf^2=vi^2+2ad
0 =70-2a(70)
70=-140a
a=-.5 mile per hour^2

Answer 2

Stopping at the store
x=x0+v0t+1/2 at^2
60=0+70+1/2(-.5)t^2

Find t

Answer 3

at what rate is he slowing down continuously? is he losing 1 m/h every minute? 5 m/h? 10?

Answer 4

I already found that, it is -0.5

Answer 5

Continuously - if he is d miles away, then he is traveling at d miles per hour.

Answer 6

I am confident that I did it right

Answer 7

I was thinking the solution involved an integral. Let T be the total time, then
\[T = \int_1^{70} f(t) dt\]
question is what's f(t) ?

Answer 8

Assuming linear speed: d = rt then t = d/r.
so I think f(t) = distance/rate.

Answer 9

Well we know that it is constant accelerationV=\[\int\limits -a \, dt\], to find velocity we integrate
we get
=-at+V0
To get position we integrate again
X=\[\int\limits -\text{at}+V\text{ }\text{dt}\]
We get
-1/2 at^2+Vt+X0

Answer 10

Wouldn't this give the total change in position?

Answer 11

Yes, but integration find us the forumla which we can use to solve problem like this

Answer 12

hum... the solution to part a is ln(70).

Answer 13

I think what's happening is the following.
D = RT
He is going d mph when he is d miles from home. How long does it take him to go the first mile when he is going 70mph? 1/70 h
Of course this changes continuously, but the idea is that it will always take him 1/d h to go 1 mile when he is d miles from home. Thus
t = 1/d so d = 1/t. And let the change in time be dt. so we get
\[\int_1^{70} \frac{dt}{t}\]
Which would get me to the solution.

Answer 14

May be someone can explain this a little better.

Answer 15

imranmeah91 -Since velocity is changing continuously, we definitely have acceleration. But how would you know that acceleration is a constant, and not some other function of time?

Answer 16

I assumed that acceleration is constant