A man starts walking north at 2 ft/s from a point p. five minutes later a woman starts walking south at 4 ft/s from a point 500
ft due east of p. at what rate are the people moving apart 15 minutes after the woman starts walking? (round your answer to two decimal places.)
1 answer:
Refer to the diagram shown below.
After 5 minutes (300 seconds):
The man travels north by (2 ft/s)*(300 s) = 600 ft
The woman, located at q, 500 east of p, begins walking south at 4 ft/s.
The distance separating them is
d₁ = √(600² + 500²) = 781.025 ft
After 20 minutes:
The man has traveled for 20 minutes (1200 s).
The woman has traveled for 15 minutes (900 s).
The man has moved (2 ft/s)*(1200 s) = 2400 ft north of p.
The woman has moved (4 ft/s)*(900 s) = 3600 ft south of q.
The distance separating them is
d₂ = √(6000² + 500²) = 6020.8 ft
The separation from d₁ to d₂ occurs in 15 minutes (900s).
Therefore the rate of separation is
Rate = (d₂ - d₁ ft)/(900 s) = (6020.8 - 781.025)/900 = 5.822 ft/s
or
Rate = (5.822 ft/s)*(60 s/min) = 349.32 ft/min
Answer: 349.32 ft/min (or 5.82 ft/s)
After 5 minutes (300 seconds):
The man travels north by (2 ft/s)*(300 s) = 600 ft
The woman, located at q, 500 east of p, begins walking south at 4 ft/s.
The distance separating them is
d₁ = √(600² + 500²) = 781.025 ft
After 20 minutes:
The man has traveled for 20 minutes (1200 s).
The woman has traveled for 15 minutes (900 s).
The man has moved (2 ft/s)*(1200 s) = 2400 ft north of p.
The woman has moved (4 ft/s)*(900 s) = 3600 ft south of q.
The distance separating them is
d₂ = √(6000² + 500²) = 6020.8 ft
The separation from d₁ to d₂ occurs in 15 minutes (900s).
Therefore the rate of separation is
Rate = (d₂ - d₁ ft)/(900 s) = (6020.8 - 781.025)/900 = 5.822 ft/s
or
Rate = (5.822 ft/s)*(60 s/min) = 349.32 ft/min
Answer: 349.32 ft/min (or 5.82 ft/s)
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Answer:
The automobile is running at speed of 23.806 meters per second.
Explanation:
From Kinematic we remember that acceleration () can be defined by this ordinary differential equation in terms of distance:
(1)
Where:
- Speed of the automobile, measured in meters per second.
- Distance travelled by the automobile, measured in meters.
If we know that , then the equation of acceleration is:
But distance covered by the vehicle is defined by the following formula:
(2)
Where:
- Arc angle, measured in radians.
- Radius, measured in radians.
Then, we expand (1) by means of this result:
And finally we get the following third order polynomial:
(3)
If we know that ,
and
, then the polynomial becomes into this:
(3b)
This polynomial can be solved analytically by Cardano's Method or by numerical methods. The roots of the polynomial are, respectivelly:
,
,
,
Both first and fourth roots are physically reasonable solution, but the latter represents the angle where automobile begins to skid <em>first</em>. Then, the automobile begins to skid at an angle of 1.563 radians relative to x axis.
The distance travelled by the automobile is: (,
)
Lastly, the speed of the automobile at this location is: ()
(4)
The automobile is running at speed of 23.806 meters per second.