A speeder traveling at a constant speed of 125km/h races past a billboard. A patrol car pursues from rest with constant acceleration of (8.0km/h)/s until it reaches its maximum speed of 190km/h, which it maintains until it catches up with the speeder. (a) How long does it take the patrol car to catch the speeder if it starts moving just as the speeder passes? (How far does each car travel?
125 km/h * 1h/3600s * t s = 1/2 (8km/hr/s * 1hr/3600s)(t sec)^2
The scale factor of 1h/3600s cancels out, and we have
125t = 4t^2
t = 125/4
a car starts from rest and travels with a constant acceleration at a distance of 500 meters. find its velocity at after 30 secs.express your answer in miles/hr in scientific notation
To find the time it takes for the patrol car to catch up with the speeder, we need to determine the time it takes for the patrol car to reach its maximum speed, and then the time it takes for the patrol car to cover the same distance as the speeder.
Step 1: Finding the time it takes for the patrol car to reach its maximum speed:
The acceleration of the patrol car is given as (8.0km/h)/s. We can convert this to m/s by dividing by 3.6 (1 km/h = 1/3.6 m/s).
acceleration = (8.0 km/h)/s = (8/3.6) m/s^2 = 20/9 m/s^2
The maximum speed of the patrol car is given as 190 km/h, which is also converted to m/s:
maximum speed = 190 km/h = (190/3.6) m/s = 475/9 m/s
To find the time it takes for the patrol car to reach its maximum speed, we can use the formula:
acceleration = (change in velocity) / (time)
time = (change in velocity) / acceleration
Using the values given:
time = (475/9) m/s / (20/9) m/s^2
time = (475/9)/(20/9)
time = 475/20
time = 23.75 s
Step 2: Finding the distance each car travels:
The speeder is traveling at a constant speed of 125 km/h. We need to convert this to m/s:
speeder's speed = 125 km/h = (125/3.6) m/s = 125/3.6 m/s
The distance traveled by each car can be calculated using the formula: distance = speed x time
Distance traveled by the patrol car:
distance = (475/9) m/s x 23.75 s
Distance traveled by the speeder:
distance = (125/3.6) m/s x 23.75 s
Simplifying the calculations will give you the final answers.
To find the time it takes for the patrol car to catch up with the speeder, we need to determine the distance each car travels.
Let's break down the problem into three parts:
Part 1: Determining the acceleration time for the patrol car
The patrol car starts from rest and accelerates at a constant rate until it reaches its maximum speed. We need to find the time it takes for the patrol car to reach its maximum speed.
Using the formula: v = u + at
Where:
v is the final velocity (190 km/h),
u is the initial velocity (0 km/h),
a is the acceleration ((8.0 km/h)/s),
t is the time we want to find.
Rearranging the formula to solve for t, we get:
t = (v - u) / a
Substituting the values, we have:
t = (190 km/h - 0 km/h) / (8.0 km/h)/s
Converting the acceleration from km/h/s to km/h² for consistency:
t = (190 km/h) / (8.0 km/h²)
Calculating:
t ≈ 23.75 s
So, it takes the patrol car approximately 23.75 seconds to reach its maximum speed.
Part 2: Determining the distance covered by the patrol car during acceleration
Using the formula: s = ut + 0.5at²
Where:
s is the distance we want to find,
u is the initial velocity (0 km/h),
t is the time of acceleration (23.75 s),
a is the acceleration ((8.0 km/h)/s).
Rearranging the formula to solve for s, we get:
s = ut + 0.5at²
Substituting the values, we have:
s = (0 km/h)(23.75 s) + 0.5((8.0 km/h)/s)(23.75 s)²
Calculating:
s ≈ 564.06 km
So, the patrol car covers approximately 564.06 kilometers during the acceleration phase.
Part 3: Finding the time and distance when the patrol car catches the speeder
Since the patrol car starts moving just as the speeder passes the billboard, the speeder has a head start of 564.06 kilometers. To catch up with the speeder, the patrol car needs to cover this distance at its constant maximum speed.
Using the formula: t = d / v
Where:
t is the time we want to find,
d is the distance (564.06 km),
v is the velocity (190 km/h).
Substituting the values, we have:
t = (564.06 km) / (190 km/h)
Calculating:
t ≈ 2.97 h
So, it takes the patrol car approximately 2.97 hours (or 2 hours and 58 minutes) to catch up with the speeder.
In summary:
- The patrol car takes approximately 23.75 seconds to reach its maximum speed.
- During this time, the patrol car covers approximately 564.06 kilometers.
- It then takes approximately 2.97 hours to catch up with the speeder.