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Example Questions
Example Question #1 : Motion In Two Dimensions
A rock is tossed horizontally off the edge of a cliff with a velocity of . How long will it take to reach a total speed of
?
This question requires an understanding of motion in two dimensions. The most important concept in this question is that the motion in each dimension is independent. Since the rock's initial velocity is purely in the horizontal direction, the initial velocity has no impact on the vertical velocity at any point. Likewise, the vertical acceleration has no impact on the horizontal speed. The rock will travel at in the horizontal direction throughout its entire trajectory.
To solve this problem, the first step is to set up an expression for the horizontal and vertical velocities as a function of time:
Thus,
Substituting for
and solving for
yields
.
Example Question #1 : Motion In Two Dimensions
A projectile reaches it max height in It has a horizontal velocity of
. What is the speed at which it is launched?
We first must find the initial vertical velocity () using:
We know that , since in 2 dimensions the vertical velocity at its max height is equal to zero.
we also know that because that information is given, and that
So plugging in what we know:
Knowing that is constant and equals
, we can use the pythagorean theorem to determine the Resultant initial Velocity:
Example Question #1 : Motion
If an object strikes the ground at what height was it dropped from?
First let's make a table of what we know:
because it has zero velocity right when it is dropped.
It should also be noted it is simpler to define things so that the initial height is zero, so we don't have to deal with a bunch of negative numbers.
Also, since it is dropped vertically and we are only interested in the height it is dropped from we aren't interested in any information regarding the projectile's horizontal motion.
Let's use the equation:
since
We can solve for , then plug in what we know:
This is our final answer.
Example Question #1 : Motion In Two Dimensions
A projectile is launched vertically upwards at . At what time does it reach its max height?
First let's write down the information we're given:
, since at a projectile's max height its vertical velocity equals zero.
Let's use the equation
to solve for the time it takes the projectile to reach its max height.
Solving for :
Now lets plug in what we know and calculate the time:
, which is our final answer.
Example Question #1 : Motion
A projectile is launched vertically upwards at . How long will it be in the air?
First let's write down the information we're given:
, since at a projectile's max height its vertical velocity equals zero.
Let's use the following equation to solve for the time it takes the projectile to reach its max height:
Solving for :
Now lets plug in what we know and calculate the time:
Since the projectile takes twice as long to land as its does to reach its max height we simply multiple the we found by
, which gives us
.
Example Question #2 : Motion
If a particle reaches its max height in , what is its range if it is launched at a speed of
that remains constant throughout its flight at angle of
?
First let's write out the information we are given:
We know that in order to obtain the time it takes a projectile to hit the ground we just multiply by
, since the time it takes to reach the max height is half of the total time the project is in the air, so
Now we can use the equation
Since we are only concerned with the particle's motion in the horizontal direction, and we know that the horizontal velocity is constant . We also know
since that is the definition of the initial position.
This gives us:
Let's plug in what we have:
. This is the range and our final answer. The reason why we can just use
as the horizontal velocity is because the project is launched at a
° angle.
Example Question #1 : Motion In Two Dimensions
A cannon is being shot from the ground. You want to shoot that cannon as far as possible. At what angle should the cannon be shot?
30 degrees
90 degrees
50 degrees
45 degrees
60 degrees
45 degrees
There are many explanations for this. First you could simply insert a velocity at all of these angles and see which ends up with the greatest change in horizontal distance. You also could use basic calculus and solve for the greatest theta.
Example Question #1 : Motion
A ball is thrown off a building at a speed of and at
to the horizontal. If the building is
tall, how far from the base of the building will the ball land?
This is a projectile motion problem. The problem states that it was thrown off the building at an angle of 30 degrees to the horizontal. So before we can find time, we need to find the horizontal and vertical parts of this velocity.
Since and
It follows that
Now, before we can find far it went, we need to find how long it was in the air. We need to solve for time.
With the equation we can find t.
X is the initial height which in this case is the tall building. V is the initial vertical velocity
and
Plugging all that in and solving yields
Knowing that the time the ball is in the air is and the constant horizontal velocity is
, we can plug in these known values into the simple distance formula to solve for distance.
Example Question #1 : Harmonic Motion
A pendulum is made up of a small mass that hangs on the end of a
long string of negligible mass. The pendulum is displaced by
and allowed to undergo harmonic motion. What is the angular frequency of the resulting motion?
The angular frequency of a simple pendulum is , where
is the length of the pendulum.
Example Question #1 : Motion
Which of the following is not an example of simple harmonic motion?
A book falling to the ground
A child swinging on a swing set
The mass on a pendulum moving back and forth
A plucked string vibrating on a guitar
A book falling to the ground
For this question, we need to recall what simple harmonic motion is. Remember that it is a periodic motion where the restoring force depends on the displacement of the object undergoing these motions. So to answer this question, we need to keep this idea in mind and see which example doesn't match up.
A mass on a pendulum moving back and forth is clearly an example of simple harmonic motion. As the mass moves further from the center in either direction, it experiences a greater and greater force in the opposite direction.
A child swinging on a swing set is another correct example. This situation is analogous to the mass on a pendulum swinging back and forth.
A vibrating guitar string is yet another example of simple harmonic motion. After it is plucked, the string oscillates back and forth.
Finally, a book falling to the ground does not represent harmonic motion. Once the book is released from rest, we intuitively know that it will fall to the ground and will then stay there; in no way is there any periodic motion.
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