a car attempts to stop and slides 25.5m while accelerating at -4m/s^2. The car's initial velocity was 20m/s. Find the time the slide took. Note that the final velocity is not known and that you need a quadratic solution.
I know the quadratic formula, but I'm not sure how to plud in the values; could sombody please calrify. Thank you in advance. :)
To find the time it took for the car to slide, we can use the following kinematic equation:
Δx = v_i * t + (1/2) * a * t^2
where:
Δx = displacement (slide distance) = -25.5 m (negative because it's a slide)
v_i = initial velocity = 20 m/s
a = acceleration = -4 m/s^2
t = time
Plugging in the given values into the equation, we get:
-25.5 = (20 * t) + (1/2) * (-4) * t^2
Simplifying the equation, we have:
-25.5 = 20t - 2t^2
Rearranging the equation to have all terms on one side:
2t^2 - 20t - 25.5 = 0
Now we have a quadratic equation in the form of at^2 + bt + c = 0, where:
a = 2
b = -20
c = -25.5
We can use the quadratic formula to solve for t. The quadratic formula is:
t = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the values into the formula:
t = (-(-20) ± √((-20)^2 - 4(2)(-25.5))) / (2(2))
Simplifying further:
t = (20 ± √(400 + 204)) / 4
t = (20 ± √604) / 4
Now we need to calculate the square root of 604.
Using a calculator, we find:
√604 ≈ 24.598
So, the equation becomes:
t = (20 ± 24.598) / 4
Now we have two possibilities for t:
1. t = (20 + 24.598) / 4 = 11.65
2. t = (20 - 24.598) / 4 = -1.65
Since time cannot be negative in this scenario, we discard the negative value. Therefore, the time it took for the car to slide is approximately 11.65 seconds.