How does changing the time affect work? How about power?
1 answer:
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Answer: 42.49
Explanation:
To solve this, we need to keep in mind the following:
While the sphere hangs it is under the effect of gravity. It is creating a Angle of 90° taking the roof as a reference.
Gravity can be noted as a Acceleration Vector. The magnitud for Earth's Gravity is a constant: 9.81
The acceleration of the Van will affect the sphere also, but this accelaration will be on the X-axis and perpendicular to the gravity. Because this two vectors are taking action under the sphere they will create a angle. This angle can be measured as a relation of the two magnitudes.
Tangent (∅) = Opossite Side / Adyacent Side
By trigonometry, we know the previous formula. This formula allows us to find the Tangent of a angle as a relation between the two perpendiculars magnitudes. In this case the Opossite Side will be the Gravity Accelaration, while the Adyancent Side is the Van's Acceleration.
(1) Tangent (∅) = Gravity's Acceleration (G) / Van's Acceleration (Va)
Searching for the Va in (1)
Va = G/Tan(∅)
Where ∅ in this case is equal to 13.0°
Va = 9.81 / Tan(13.0°)
Va = 42.49
The vans acceleration need to be 42.49 to create an angle of 13° with the Van's Roof
The complete text of the problem is:
<em>"Traumatic brain injury such as concussion results when the head undergoes a very large acceleration. Generally, an acceleration less than 800 m/s2 lasting for any length of time will not cause injury, whereas an acceleration greater than 1000 m/s2 lasting for at least 1 ms will cause injury. Suppose a small child rolls off a bed that is 0.43 m above the floor. If the floor is hardwood, the child's head is brought to rest in approximately 1.8 mm. If the floor is carpeted, this stopping distance is increased to about 1.1 cm. Calculate the magnitude and duration of the deceleration in both cases, to determine the risk of injury. Assume the child remains horizontal during the fall to the floor. Note that a more complicated fall could result in a head velocity greater or less than the speed you calculate. "</em>
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<u>Solution:</u>
1) Acceleration: on the hardwood floor,
on the carpeted floor
First of all, we need to calculate the speed of the child just before he hits the floor. This can be done by using the equation
where
v is the final speed
u = 0 is the initial speed (the child starts from rest)
a = g = 9.8 m/s^2 is the acceleration of gravity
d = 0.43 m is the distance covered by the child as he falls from the bed
Solving for v,
Now we can analyze the moment of the collision. The child hits the floor with an initial speed of v = 2.9 m/s, and he comes to a stop, so the final speed is v' = 0. If the floor is hardwood, the stopping distance is
So we can find the acceleration by using again the equation
Solving for a,
For the carpeted floor instead,
therefore the acceleration is
2) Duration: 1.24 ms for the hardwood floor, 7.59 ms for the carpeted floor
We can find the duration of the collision in both cases by using the equation of the acceleration
where
v' = 0
v = 2.9 m/s
For the hardwood floor,
So the duration of the collision is
For the carpeted floor,
So the duration of the collision is
We can now comment the results using the initial statement of the problem:
"Generally an acceleration less than 800 m/s2 lasting for any length of time will not cause injury, whereas an acceleration greater than 1,000 m/s2 lasting for at least 1ms will cause injury"
Therefore, the fall on the hardwood floor can result in injury (since the acceleration is greater than 1,000 m/s2 for more than 1 ms), while the fall on the carpeted floor is not dangerous (much less than 1000 m/s^2).