A dog walking to the right at 1.5 m/s spies a cat ahead, and begins chasing the cat with a constant acceleration of 12 m/s2. What is the velocity of the dog after running for 3.0 m?

Answer 1

We know that

v(t) = vo + a*t

where vo is the initial velocity, v(t) is the velocity at time t, and a is the acceleration.

For the displacement under constant acceleration (as given here) we can write

<v> = (vo + v(t)) / 2

where <v> is the average velocity between 0 and t. By definition, the average velocity is also given as

<v> = (x(t) - xo) / t

where x(t) is the position at time t and x0 is the position at time = 0. This simply states that the average velocity is the displacement (x(t) - xo) divided by the time interval (t).

Rearranging,

x(t) - x0 = <v> * t

Note from the first equation above we have:

t = (v(t) - vo) / a

so we can write

x(t) - xo = <v> * (v(t) - vo) / a = (v(t) + vo) * (v(t) - vo) / (2 * a) = (v^2 - v0^2) / (2 * a)

or

v^2 = vo^2 + 2 * a * (x-x0)

v = sqrt(vo^2 + 2 * a * (x-x0))

Now we can find the dog's velocity after he has travelled 3 meters:

vo = 1.5; a = 12; x-xo = 3

v = sqrt((1.5)^2 + 2 * 12 * 3) = 8.62 m/sec.