Braking Car-Physics Question?

Question

A car traveling at a speed of vo = 58 m/s stops smoothly (that is, its deceleration is constant) over a distance of d = 111 m. a) What is its acceleration during the time it is stopping? (Be careful about the sign!)

a = m/s2
-5.07 NO
b) How long (what amount of time) does it take for the car to come to a stop?

tstop = s
c) After the car has gone 1/3 of the stopping distance, what is its speed?

v1/3 = m/s
27.76 NO

HELP: Dx = d/3
d) If the car had been going 10 m/s but had stopped in the same d = 111 m distance, what would its acceleration have been?

a(vo = 10m/s) = m/s2

Answer 1

The key to it is that if the deceleration is constant from Vo to zero, then the average speed over the interval is half the initial velocity, or Vo/2.

That makes the time = distance / average speed, or 222/58,
or about 3.83 seconds.

The acceleration is the change in velocity divided by the time.
-58/3.83= -15.15 m/s2

Using distance=(1/2) x (acceleration) x (time) x (time), the time it reaches 1/3 the stopping distance is sqrt(111 x 2/(3 x 15.15)) or 2.21 seconds.

Hold the phone...

After sleeping on it, that's for positive accelerations and zero initial velocity!

I'll get back to you on the more general case...






You still here? 'Glad you posted to such a quiet corner of the forum...

The time to go one third the distance is buried in the expression:

d/3 = (1/2)*acceleration*time*time + Vo* time

Keep in mind that in our case the acceleration is a negative number.

Since it's quadratic in time, the solution looks a bit complicated.

if "a" is acceleration, "t" is time, and "d" is distance:

t= (-Vo + sqrt(Vo*Vo - 2(a*d/3)))/a

...but it works out to about 0.70 seconds.

(There is another solution using the negative of the square root-- but that corresponds to the acceleration continuing as the car stops, reverses direction, and RETURNS to the d/3 position...)

The speed at that point is Vo + (acceleration x time)

or 58 + (-15.15 x 0.70) or 47.36 m/s

Similarly, had the car been going 10 m/s, the average speed would have been 5 m/s, the time 22.2 seconds, and the acceleration would be -0.45 m/s2.

Thanks for the review :)