Preparation problems
5-3. A small plane starts down the runway with
acceleration 7.2 m/s2. If the force provided by its engine is
1.1×104 N, what is the plane's mass?
Solution:
This problem is a straightforward application of Newton's
Second Law, so we start by writing down that equation:
We are given a force and an acceleration, both of which are in the same
direction. This means we can consider the problem to only occur in one
dimension and drop the vector signs from our force and acceleration.
This makes Newton's Second Law become:
5-10. By how much does the force required to stop a car
increase if (a) the stopping time is halved and (b) the
stopping distance is halved?
Solution:
We will need to connect the force applied to the car to the acceleration
experienced by the car. To do this, we need to recall Newton's Second
Law:
Since we are acting in only one dimension (let's say the x-direction),
we can drop the vector notation and write
Now we can look at our kinematic equations to solve the rest of the
problem.
(a) We want to look at the second kinematic equation,
remembering that since we are stopping, the final velocity is zero:
where F1 is the force required to stop the car in time. What happens
to the force if we cut the time in half? Well we would start from the
same velocity, and so we can put in the new force neccessary to stop the
car in half the time, F2, and find out how big it must be
(b) Now we want to look at the third kinematic equation,
remembering that since we are stopping, the final velocity is zero. Our
starting position, x0, we will also assume to be zero.
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vx02 + 2 |
æ
ç
è
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Fa
m
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ö
÷
ø
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x |
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where Fa is the force required to stop the car in time. What happens
to the force if we halve the stopping distance? Well we would start from the
same velocity, and so we can put in the new force neccessary to stop the
car in half the distance, Fb, and find out how big it must be
5-14. As a function of time, the velocity of an object
of mass m is given by
v = bt2 i +(ct+d) j, where b, c, and d are constants with
appropriate units. What is the force acting on the object, as a
function of time?
Solution:
This problem is another question about Newton's
Second Law, so we start by writing down that equation:
Since we are only given the velocity of the object, we must find its
acceleration by recallng
Remembering that when we take the time derivative of a vector function,
we do so independantly for each component, we can find the acceleration
of the object:
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d
dt
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é
ë
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bt2 |
^
i
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+ (ct+d) |
^
j
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ù
û
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d
dt
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(bt2) |
^
i
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+ |
d
dt
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(ct+d) |
^
j
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Now we can use Equation (1) to find the force acting on our
object:
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m |
é
ë
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(2bt) |
^
i
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+ (c) |
^
j
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ù
û
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Solutions translated from TEX by TTH, version 1.57.
