While driving down Zelzah on your way to school you notice that campus police are hiding at the entrance to the high school and you are traveling at 30 m/s, you only have 20 meters to reach the speed limit of 35 mi/hr. What acceleration would you experience in order to accomplish this?
35 mi/hr = 15.65 m/s
30 + at = 15.65
t = -14.35/a
30t + 1/2 at^2 = 20
30(-14.35/a) + 1/2 a (-14.35/a)^2 = 20
a = -16.377 m/s^2
To determine the required acceleration, we need to convert the speed from meters per second (m/s) to miles per hour (mi/hr) since the speed limit is given in miles per hour.
1 m/s is approximately equal to 2.237 mi/hr.
So, the initial speed in mi/hr is calculated as follows:
Initial speed = 30 m/s * 2.237 mi/hr/m = 67.11 mi/hr
To reach the speed limit of 35 mi/hr, the change in speed required is:
Change in speed = 35 mi/hr - 67.11 mi/hr = -32.11 mi/hr
Since we only have 20 meters to reach the speed limit, we need to convert this distance to miles as well.
1 meter is approximately equal to 0.000621371 mi.
So, the distance in mi is calculated as follows:
Distance = 20 m * 0.000621371 mi/m = 0.012427 mi
Now, we can calculate the required acceleration using the following formula:
Acceleration = (Change in speed) / (Time to change speed)
Given that distance is related to acceleration and time through the equation:
Distance = (1/2) * acceleration * time^2
Rearranging the equation, we get:
Time = sqrt((2 * distance) / acceleration)
Substituting the known values:
Time = sqrt((2 * 0.012427 mi) / acceleration)
Now, we can substitute this time expression into the acceleration equation:
Acceleration = (Change in speed) / (sqrt((2 * 0.012427 mi) / acceleration))
Simplifying the equation, we get:
Acceleration^2 = (Change in speed)^2 / (2 * 0.012427 mi)
Taking the square root of both sides, we get:
Acceleration = sqrt((Change in speed)^2 / (2 * 0.012427 mi))
Finally, we can substitute the values and calculate the acceleration.
Acceleration = sqrt((-32.11 mi/hr)^2 / (2 * 0.012427 mi))
Acceleration ≈ 68.67 mi/hr^2
So, you would need to experience an acceleration of approximately 68.67 mi/hr^2 in order to reach the speed limit of 35 mi/hr within the given distance of 20 meters.
To determine the acceleration required to reach the speed limit, we can use the following equation:
v = u + at
Where:
v = final velocity (35 mi/hr)
u = initial velocity (30 m/s)
a = acceleration (unknown)
t = time
First, we need to convert the final velocity to m/s:
35 mi/hr * 1.60934 km/mi * 1000 m/km * 1 hr/3600 s ≈ 15.65 m/s
Now we can substitute the values into the equation:
15.65 m/s = 30 m/s + a(t)
Since the initial velocity is already given and we are looking for acceleration, we can simplify the equation:
a(t) = 15.65 m/s - 30 m/s
a(t) = -14.35 m/s
Now, we know that the time to cover the remaining 20 meters (distance) is:
20 meters / 30 m/s = 2/3 second
We can substitute this value back into the equation:
-14.35 m/s = a(2/3 s)
Finally, we can solve for acceleration (a):
a = (-14.35 m/s) / (2/3 s)
a ≈ -21.53 m/s²
Therefore, to reach the speed limit of 35 mi/hr within a distance of 20 meters, you would experience an acceleration of approximately -21.53 m/s². Note that the negative sign indicates that the acceleration is in the opposite direction of your initial velocity.