A speeder traveling at a constant speed of

102 km/h races past a billboard. A patrol car
pursues from rest with constant acceleration
of (9 km/h)/s until it reaches its maximum
speed of 146 km/h, which it maintains until it
catches up with the speeder.
How long does it take the patrol car to
catch the speeder if it starts moving just as
the speeder passes?
Answer in units of s

To find the time it takes for the patrol car to catch up with the speeder, we can use the equation of motion for constant acceleration:

v = u + at

where:
v = final velocity
u = initial velocity
a = acceleration
t = time

First, let's convert the velocities from km/h to m/s to make the units consistent:

Speeder's velocity: 102 km/h = (102 * 1000) / (60 * 60) = 28.33 m/s
Maximum velocity of the patrol car: 146 km/h = (146 * 1000) / (60 * 60) = 40.56 m/s

Since the speeder is already moving at a constant speed of 28.33 m/s and the patrol car starts from rest, the initial velocity of the patrol car (u) is 0 m/s.

Now, we need to find the time it takes for the patrol car to reach its maximum velocity. We can use the equation:

v = u + at

Rearranging the equation to solve for time (t):

t = (v - u) / a

For the patrol car, v = 40.56 m/s, u = 0 m/s, and a = 9 km/h/s = (9 * 1000) / (60 * 60) = 2.5 m/s². Plugging these values into the equation, we get:

t = (40.56 - 0) / 2.5 = 16.22 seconds (rounded to two decimal places)

Therefore, it takes the patrol car approximately 16.22 seconds to reach its maximum speed.

Now, we need to find the time it takes for the patrol car to catch up with the speeder. Since both vehicles are moving at a constant speed after the patrol car reaches its maximum velocity, the time taken to catch up will be the same as the time taken to cover the distance between them.

To find this, we can use the formula for calculating distance traveled with a constant speed:

distance = speed × time

The distance traveled by the patrol car (d_pc) is the same as the distance traveled by the speeder (d_s), so we can equate the two equations:

speed_pc × time = speed_s × time

If we rearrange the equation to solve for time, we get:

time = distance / (speed_pc - speed_s)

The distance between them is not given in the problem statement, but since the patrol car starts moving just as the speeder passes a billboard, we can assume that the initial distance between them is zero.

Plugging in these values, we get:

time = 0 / (40.56 - 28.33) = 0 / 12.23 = 0 seconds

Therefore, it takes 0 seconds for the patrol car to catch the speeder if it starts moving just as the speeder passes.

Answer: 0 seconds.