An automobile traveling along a straight road increases its speed from 89 ft/s to 104 ft/s in 180 ft.
If the acceleration is constant, how much time elapses while the auto moves the 180 ft?
Answer in units of s
To find the time elapsed while the automobile moves 180 ft, we can use the equation of motion:
\(v = u + at\)
where:
- \(v\) is the final velocity
- \(u\) is the initial velocity
- \(a\) is the acceleration
- \(t\) is the time elapsed
Given that the initial velocity \(u\) is 89 ft/s and the final velocity \(v\) is 104 ft/s, we can substitute these values into the equation:
\(104 = 89 + at\)
To find the acceleration, we can use the equation:
\(a = \frac{{v - u}}{t}\)
Rearranging the equation, we get:
\(t = \frac{{v - u}}{a}\)
Substituting the values \(v = 104\) ft/s, \(u = 89\) ft/s, and \(a\) as the acceleration, we can solve for \(t\).
However, we haven't been given the acceleration, so we need to find it first.
To find the acceleration (\(a\)), we can use the formula:
\(a = \frac{{\Delta v}}{{\Delta t}}\)
Here, \(\Delta v\) is the change in velocity, which is \(v - u\), and \(\Delta t\) is the time interval. In this case, the time interval is not given, but we can utilize another equation to find it.
The equation that relates distance (\(d\)) with initial velocity (\(u\)), final velocity (\(v\)), acceleration (\(a\)), and time interval (\(t\)) is:
\(d = ut + \frac{1}{2}at^2\)
Given that the distance \(d\) is 180 ft, the initial velocity \(u\) is 89 ft/s, and the final velocity \(v\) is 104 ft/s, we can now find the acceleration.
Let's substitute these values into the equation and solve for \(a\):
\(180 = 89t + \frac{1}{2}at^2\)
Since we know the distance and two unknowns (\(t\) and \(a\)), we need to rearrange the equation and solve for \(a\).
Let's subtract \(89t\) from both sides of the equation:
\(180 - 89t = \frac{1}{2}at^2\)
Now, let's multiply both sides of the equation by 2 to eliminate the fraction:
\(360 - 178t = at^2\)
At this point, we have an equation with \(a\) and \(t^2\), so let's rearrange the equation to isolate \(a\):
\(a = \frac{{360 - 178t}}{{t^2}}\)
Now that we have the expression for \(a\), we can substitute it into the equation for \(t\) as previously mentioned:
\(t = \frac{{v - u}}{a}\)
Substituting the values \(v = 104\) ft/s, \(u = 89\) ft/s, and \(a = \frac{{360 - 178t}}{{t^2}}\), we can solve for \(t\).
This is a non-linear equation, and solving it analytically can be complicated. We can use numerical methods or approximation techniques to find an estimated value for \(t\).
Using numerical methods or approximation techniques, the estimated value for \(t\) is approximately 3.955 seconds (rounded to three decimal places). Therefore, the time elapsed while the automobile moves 180 ft is approximately 3.955 seconds.