Kent Belate is driving down I-294 at 27.5 m/s. He accelerates at a uniform rate of 5.27 m/s/s over a distance of 50.7 m. How much time is required to cover this distance?

Answer 1

To determine the time required to cover the distance, we can use the formula:

distance = (initial velocity * time) + (0.5 * acceleration * time^2)

Rearranging the formula, we get:

time = (distance - (0.5 * acceleration * time^2)) / initial velocity

Given:
initial velocity (v0) = 27.5 m/s
acceleration (a) = 5.27 m/s^2
distance (d) = 50.7 m

We can substitute these values into the formula and solve for time:

time = (50.7 - (0.5 * 5.27 * time^2)) / 27.5

Multiplying through by 27.5, we get:

27.5 * time = 50.7 - (0.5 * 5.27 * time^2)

Expanding the right side of the equation, we get:

27.5 * time = 50.7 - 2.635 * time^2

Rearranging the equation, we get:

2.635 * time^2 + 27.5 * time - 50.7 = 0

This is a quadratic equation. We can solve it by using the quadratic formula:

time = (-b ± sqrt(b^2 - 4ac)) / 2a

Plugging in the values from our equation, we get:

time = (-(27.5) ± sqrt((27.5)^2 - 4(2.635)(-50.7))) / (2 * 2.635)

Simplifying this equation, we get:

time = (-27.5 ± sqrt(756.25 + 618.0385)) / 5.27

time = (-27.5 ± sqrt(1374.2885)) / 5.27

time = (-27.5 ± 37.08) / 5.27

Using the positive value for time:

time = (-27.5 + 37.08) / 5.27

time = 9.58 / 5.27

time = 1.82 seconds

Therefore, it takes approximately 1.82 seconds to cover the distance.