The brakes on your automobile are capable of slowing down your car at a rate of 4.2 m/s^2.
a) If you are going 145 km/h and suddenly see a state trooper, what is the minimum time in which you can get your car under the 90 km/h speed limit?
b) Graph x vs. t and v vs. t for such a deceleration.
I have tried this problem several different ways, but I still cannot get it. I'm pretty confused with this and it's all new to me, so help would be appreciated.
a) 145 km/h = 40.28 m/s
You want to reduce the speed to 90 km/h = 25.00 m/s
The time T that it takes to decelerate to the lower speed at the fasted deceleration rate, a = 4.2 m/s^2, is given by
25 = 40.28 - a T
T = (40.28-25)/a = 3.64 s
b) Graph V = 40.28 - 4.2 t (in m/s)
If you want to graph V(t) with units of km/h, plot
V (km/h) = 145 - 15.12 t
a) To determine the minimum time it takes to get your car under the 90 km/h speed limit, we need to calculate the deceleration required. The deceleration is given as 4.2 m/s^2, which is equivalent to a change in velocity of -4.2 m/s every second.
First, let's convert the initial velocity and the speed limit from km/h to m/s to ensure consistent units.
Initial velocity: 145 km/h = (145 * 1000) m/3600 s = 40.278 m/s
Speed limit: 90 km/h = (90 * 1000) m/3600 s = 25 m/s
Now, we can use the equation of motion to find the time it takes to decelerate from 40.278 m/s to 25 m/s:
v = u + at
Where:
v = final velocity (25 m/s)
u = initial velocity (40.278 m/s)
a = deceleration (-4.2 m/s^2)
t = time (unknown)
Rearranging the equation, we have:
t = (v - u) / a
Substituting in the values, we get:
t = (25 - 40.278) / (-4.2) ≈ 3.67 seconds
Therefore, the minimum time required to get the car under the 90 km/h speed limit is approximately 3.67 seconds.
b) To graph the position (x) vs. time (t) and velocity (v) vs. time (t) for such a deceleration, we need to define the initial conditions.
Let's assume the initial position x0 = 0 and the initial velocity v0 = 40.278 m/s.
For the position vs. time graph, we can use the equation of motion:
x = x0 + v0t + (1/2)at^2
Substituting the values, we have:
x = 0 + 40.278t + (1/2)(-4.2)t^2
For the velocity vs. time graph, we can use the equation:
v = v0 + at
Substituting the values, we have:
v = 40.278 - 4.2t
You can plot these equations on a graphing tool or a spreadsheet software to visualize the position and velocity changes over time. The x-axis represents time (t), the y-axis represents position (x) in the position graph, and velocity (v) in the velocity graph.
This should help you understand the problem better and guide you in solving similar problems in the future. If you have any further questions, feel free to ask!