An Air Force plane lands with a velocity of 125 m/s and accelerates at a maximum rate of -6.5 m/s^2.
1 answer:
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a) Same
b) Same
c) Same
d) Throw the ball takes longer
e) F is larger when the ball is catched
Explanation:
a)
The change in speed of an object is given by:
where
u is the initial velocity of the object
v is the final velocity of the object
The change in speed is basically the magnitude of the change in velocity (because velocity is a vector, while speed is a scalar, so it has no direction).
In this problem:
- In situation 1 (pitcher throwing the ball), the initial velocity is
u = 0 (because the ball starts from rest)
while the final velocity is v, so the change in speed is
- In situation 2 (catcher receiving the ball), the initial velocity is now
u = v
while the final velocity is now zero (ball coming to rest), so the change in speed is
Which means that the two situations have same change in speed.
b)
The change in momentum of an object is given by
where
m is the mass of the object
is the change in velocity
If we want to compare only the magnitude of the change in momentum of the object, then it is given by
- In situation 1 (pitcher throwing the ball), the change in momentum is
- In situation 2 (catcher receiving the ball), the change in momentum is
So, the magnitude of the change in momentum is the same (but the direction is opposite)
c)
The impulse exerted on an object is equal to the change in momentum of the object:
where
I is the impulse
is the change in momentum
As we saw in part b), the change in momentum of the ball in the two situations is the same, therefore the impulse exerted on the ball will also be the same, in magnitude.
However, the direction will be opposite, as the change in momentum has opposite direction in the two situations.
d)
To compare the time of impact in the two situations, we have to look closer into them.
- When the ball is thrown, the hand "moves together" with the ball, from back to ahead in order to give it the necessary push. We can verify therefore that the time is longer in this case.
- When the ball is cacthed, the hand remains more or less "at rest", it doesn't move much, so the collision lasts much less than the previous situation.
Therefore, we can say that the time of impact is longer when the ball is thrown, compared to when it is catched.
e)
The impulse exerted on an object can also be rewritten as the product between the force applied on the object and the time of impact:
where
I is the impulse
F is the force applied
is the time of impact
This can be rewritten as
In this problem, in the two situations,
- I (the impulse) is the same in both situations
- when the ball is thrown is larger than when it is catched
Therefore, since F is inversely proportional to , this means that the force is larger when the ball is catched.
We divide the thin rectangular sheet in small parts of height b and length dr. All these sheets are parallel to b. The infinitesimal moment of inertia of one of these small parts is
where
Now we find the moment of inertia by integrating from
to 
The moment of inertia is
(from (-a/2) to
(a/2))
where
Now we find the moment of inertia by integrating from
The moment of inertia is