A police car is traveling at a velocity of 20.0 m/s due north, when a car zooms by at a constant velocity of 44.0 m/s due north. After a reaction time 0.700 s the policeman begins to pursue the speeder with an acceleration of 6.00 m/s^2. Including the reaction time, how long does it take for the police car to catch up with the speeder?
44(t+0.7) = 20t + 3t^2
3t^2 - 24t - 30.8 = 0
I plugged this into the quadratic formula but I cannot seem to get the correct answer. Please help.
To solve the quadratic equation 3t^2 - 24t - 30.8 = 0, let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 3, b = -24, and c = -30.8. Plugging in these values, we get:
t = (-(-24) ± √((-24)^2 - 4 * 3 * (-30.8))) / (2 * 3)
t = (24 ± √(576 + 369.6)) / 6
t = (24 ± √945.6) / 6
Now let's evaluate the two cases, using both the positive and negative square root:
t1 = (24 + √945.6) / 6
t2 = (24 - √945.6) / 6
Calculating these values, we get:
t1 ≈ 6.353
t2 ≈ -0.686
Since time cannot be negative, we can discard the negative value. Therefore, the time it takes for the police car to catch up with the speeder is approximately 6.353 seconds after the reaction time.
To solve the equation 3t^2 - 24t - 30.8 = 0 using the quadratic formula, we can use the formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 3, b = -24, and c = -30.8. Substituting these values into the equation, we have:
t = (24 ± √((-24)^2 - 4 * 3 * -30.8)) / (2 * 3)
Simplifying the equation under the square root gives:
t = (24 ± √(576 + 369.6)) / 6
t = (24 ± √945.6) / 6
We can further simplify the equation under the square root:
t = (24 ± 30.802) / 6
Splitting the equation into two parts:
Case 1: t = (24 + 30.802) / 6 = 9.134
Case 2: t = (24 - 30.802) / 6 = -1.634
Since time cannot be negative, we discard Case 2. Therefore, the correct answer is t = 9.134 seconds.
Including the reaction time of 0.700 s, the total time it takes for the police car to catch up with the speeder is:
Total time = 9.134 s + 0.700 s = 9.834 seconds