Steven W. answered • 09/09/16
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{Sorry; I put this in the wrong box before}
Mayuran's solution is right on, except for one small but significant error.
Let's put this in a context that you can use to solve other kinematic problems of this type.
Kinematics deals with five kinematic quantities: initial velocity (vo), final velocity (v), displacement (x-xo), acceleration (a), and time (t). There are usually four kinematic equations given that relate these five quantities in various combinations. Each equation involves four of the five quantities.
This means that, if we want to solve for one of these quantities, we need to know at least three others. That is usually the first setup I try to make out of these kinematics questions: what do we want to find, and what do we know.
I like to start with what we want to find among the kinematic quantities, since the question usually has to state it right up front. In this case (Part (a)), it directly asks for the magnitude of the acceleration. So we want to find a.
to find: a
Then, we have to figure out which quantities we know. We have to know at least three.
We are told that, at the start of the collision, the meteorite is traveling at 500 m/s. This is the initial velocity, vo (actually, it is the initial speed, since we are not told which direction the meteorite travels in, but that is not too important here). It is good we know vo, because that is the one kinematic quantity that appears in all the kinematic equations, so we must either know it or be trying to find it to use the equations.
We can assume that, at the end of its trip, the meteorite is at rest, so its final velocity v = 0.
And we are told that the meteorite leaves a dent 20 cm deep. So we know the magnitude of its displacement over this action. It will be helpful to put this length into the same units as used in the velocity (meters). Here is where Mayuran made that small error (and one that I have NEVER made... at all... ever... ahem...). 20 cm --> 0.2 m, not 0.02 m.
So, now, we have our three kinematic quantity knowns:
know: a, v, vo
With what we want to find and what we know spelled out, we can now go to our stable of kinematic equations and choose the one that involves those four quantities. As Mayuran pointed out, it is the one that looks like:
v2 = vo2+2a(x-xo)
[this procedure of determining which kinematic to find, and which three or more are known, and then selecting the kinematic equation that includes those quantities can be applied to many kinematics problems]
Putting in our known quantities yields:
02 = (500 m/s)2+2a(0.2 m) (I have implicitly defined the meteorite's direction of motion as positive here)
You can then solve this for a.
Mayuran was exactly on about Part b, so no need to repeat that, except to make an adjustment based on the change in the value of the solution for Part (a).
One thing to note, though, is that, in this problem, there was no particular assumption made about air resistance. That is an assumption we typically make when applying kinematics to a projectile (an object whose ONLY acceleration is provided by gravity). But this is not a projectile, as we are looking at it during the time when the car is causing its acceleration.
The assumption we ARE making here is that, as with all cases where kinematic equations are used, the acceleration is one constant value. It is very likely that, in actuality, the acceleration of the meteorite is changing during the time the car is bringing it to a stop. But kinematic equations assume one constant acceleration, so that is what we solved for. We can also say that we solved for the average acceleration over the entire stopping action.
I hope this helps! If you have any other questions about this, just ask.
Let's put this in a context that you can use to solve other kinematic problems of this type.
Kinematics deals with five kinematic quantities: initial velocity (vo), final velocity (v), displacement (x-xo), acceleration (a), and time (t). There are usually four kinematic equations given that relate these five quantities in various combinations. Each equation involves four of the five quantities.
This means that, if we want to solve for one of these quantities, we need to know at least three others. That is usually the first setup I try to make out of these kinematics questions: what do we want to find, and what do we know.
I like to start with what we want to find among the kinematic quantities, since the question usually has to state it right up front. In this case (Part (a)), it directly asks for the magnitude of the acceleration. So we want to find a.
to find: a
Then, we have to figure out which quantities we know. We have to know at least three.
We are told that, at the start of the collision, the meteorite is traveling at 500 m/s. This is the initial velocity, vo (actually, it is the initial speed, since we are not told which direction the meteorite travels in, but that is not too important here). It is good we know vo, because that is the one kinematic quantity that appears in all the kinematic equations, so we must either know it or be trying to find it to use the equations.
We can assume that, at the end of its trip, the meteorite is at rest, so its final velocity v = 0.
And we are told that the meteorite leaves a dent 20 cm deep. So we know the magnitude of its displacement over this action. It will be helpful to put this length into the same units as used in the velocity (meters). Here is where Mayuran made that small error (and one that I have NEVER made... at all... ever... ahem...). 20 cm --> 0.2 m, not 0.02 m.
So, now, we have our three kinematic quantity knowns:
know: a, v, vo
With what we want to find and what we know spelled out, we can now go to our stable of kinematic equations and choose the one that involves those four quantities. As Mayuran pointed out, it is the one that looks like:
v2 = vo2+2a(x-xo)
[this procedure of determining which kinematic to find, and which three or more are known, and then selecting the kinematic equation that includes those quantities can be applied to many kinematics problems]
Putting in our known quantities yields:
02 = (500 m/s)2+2a(0.2 m) (I have implicitly defined the meteorite's direction of motion as positive here)
You can then solve this for a.
Mayuran was exactly on about Part b, so no need to repeat that, except to make an adjustment based on the change in the value of the solution for Part (a).
One thing to note, though, is that, in this problem, there was no particular assumption made about air resistance. That is an assumption we typically make when applying kinematics to a projectile (an object whose ONLY acceleration is provided by gravity). But this is not a projectile, as we are looking at it during the time when the car is causing its acceleration.
The assumption we ARE making here is that, as with all cases where kinematic equations are used, the acceleration is one constant value. It is very likely that, in actuality, the acceleration of the meteorite is changing during the time the car is bringing it to a stop. But kinematic equations assume one constant acceleration, so that is what we solved for. We can also say that we solved for the average acceleration over the entire stopping action.
I hope this helps! If you have any other questions about this, just ask.
Amy H.
Thank you
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09/09/16
Mayuran K.
09/09/16