A drag racer crosses the finish line of a 304.8 m track with a final speed of 121 m/s.
(a) Assuming constant acceleration during the race, find the racer's time and the minimum coefficient of static friction between the tires and the road.
time 1 s
minimum coefficient of friction 2
(b) If, because of bad tires or wet pavement, the acceleration were 31.1% smaller, how long would it take to finish the race?
3 s
To solve this problem, we can use the equations of motion for uniformly accelerated motion. Let's break it down step by step.
Step 1: Find the time taken to cross the finish line with a final speed of 121 m/s and a track length of 304.8 m.
We can use the following equation:
vf = vi + at
Here, vf is the final velocity (121 m/s), vi is the initial velocity (0 m/s), t is the time taken, and a is the acceleration.
Since the acceleration is constant, we can rearrange the equation to solve for time:
t = (vf - vi) / a
Given that vi = 0 m/s and vf = 121 m/s, we need to find the value of acceleration.
Step 2: Find the acceleration.
To find the acceleration, we can use another equation:
vf^2 = vi^2 + 2ad
Here, d is the distance traveled (304.8 m), vi is the initial velocity (0 m/s), vf is the final velocity (121 m/s), and a is the acceleration.
Rearranging the equation, we have:
a = (vf^2 - vi^2) / (2d)
Given that vi = 0 m/s, vf = 121 m/s, and d = 304.8 m, we can plug in these values to find the acceleration.
Step 3: Find the time.
Using the acceleration value obtained from Step 2, we can now substitute it back into the equation for time:
t = (vf - vi) / a
Plugging in the values vi = 0 m/s, vf = 121 m/s, and a into this equation will give us the time taken to cross the finish line.
(a) Using the given values, we can calculate:
vi = 0 m/s
vf = 121 m/s
d = 304.8 m
First, find the acceleration:
a = (121^2 - 0^2) / (2 * 304.8) = 60.5 m/s^2
Next, find the time:
t = (121 - 0) / 60.5 = 2 seconds
So, the time taken to cross the finish line is 2 seconds.
Now, let's move on to finding the minimum coefficient of static friction between the tires and the road.
The equation we can use is:
μs = (a / g)
Here, a is the acceleration and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Plugging in the value of acceleration from Step 2:
μs = (60.5 / 9.8) ≈ 6.17
However, the coefficient of static friction cannot be greater than 1. Therefore, the minimum coefficient of static friction is:
μs = 1
Hence, the minimum coefficient of static friction between the tires and the road is 1.
(b) Now, let's consider a decrease in acceleration by 31.1%.
Given vf = 121 m/s and d = 304.8 m, we need to find the new time taken with reduced acceleration.
First, find the new value of acceleration:
Reduced acceleration = 60.5 m/s^2 - 0.311 * 60.5 m/s^2 = 41.64 m/s^2
Next, find the time:
t = (121 - 0) / 41.64 = 2.91 seconds
So, with a decrease in acceleration by 31.1%, it would take approximately 2.91 seconds to finish the race.