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?
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.
To find the time required to cover a given distance with a uniform acceleration, we can use the following equation:
distance = initial velocity * time + 0.5 * acceleration * time^2
Given:
Initial velocity (u) = 27.5 m/s
Acceleration (a) = 5.27 m/s^2
Distance (d) = 50.7 m
Plugging these values into the equation, we get:
50.7 = 27.5 * t + 0.5 * 5.27 * t^2
Rearranging the equation, we have a quadratic equation:
0.5 * 5.27 * t^2 + 27.5 * t - 50.7 = 0
To solve for t, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
Here, a = 0.5 * 5.27, b = 27.5, and c = -50.7. Substituting these values into the quadratic formula, we can calculate t.
To find the time required to cover a distance, we can use the kinematic equation:
\[ s = ut + \frac{1}{2}at^2 \]
where:
s = distance covered (in this case, 50.7 m)
u = initial velocity (27.5 m/s)
a = acceleration (5.27 m/s/s)
t = time taken
We need to rearrange the equation to solve for t:
\[ t = \sqrt{\frac{2s}{a}} - \frac{u}{a} \]
Let's substitute the given values into the equation:
\[ t = \sqrt{\frac{2 \times 50.7}{5.27}} - \frac{27.5}{5.27} \]
Now we can calculate the value of t.