Substituting the expression for
t into the equation for the position
gives
Rearranging terms, we have
This trajectory equation is of the form
which is an equation of a parabola with coefficients
Range
From the trajectory equation we can also find the
range , or the horizontal distance traveled by the projectile. Factoring
[link] , we have
The position
y is zero for both the launch point and the impact point, since we are again considering only a flat horizontal surface. Setting
y = 0 in this equation gives solutions
x = 0, corresponding to the launch point, and
corresponding to the impact point. Using the trigonometric identity
and setting
x =
R for range, we find
Note particularly that
[link] is valid only for launch and impact on a horizontal surface. We see the range is directly proportional to the square of the initial speed
and
, and it is inversely proportional to the acceleration of gravity. Thus, on the Moon, the range would be six times greater than on Earth for the same initial velocity. Furthermore, we see from the factor
that the range is maximum at
These results are shown in
[link] . In (a) we see that the greater the initial velocity, the greater the range. In (b), we see that the range is maximum at
This is true only for conditions neglecting air resistance. If air resistance is considered, the maximum angle is somewhat smaller. It is interesting that the same range is found for two initial launch angles that sum to
The projectile launched with the smaller angle has a lower apex than the higher angle, but they both have the same range.
Comparing golf shots
A golfer finds himself in two different situations on different holes. On the second hole he is 120 m from the green and wants to hit the ball 90 m and let it run onto the green. He angles the shot low to the ground at
to the horizontal to let the ball roll after impact. On the fourth hole he is 90 m from the green and wants to let the ball drop with a minimum amount of rolling after impact. Here, he angles the shot at
to the horizontal to minimize rolling after impact. Both shots are hit and impacted on a level surface.
(a) What is the initial speed of the ball at the second hole?
(b) What is the initial speed of the ball at the fourth hole?
(c) Write the trajectory equation for both cases.
(d) Graph the trajectories.
Strategy
We see that the range equation has the initial speed and angle, so we can solve for the initial speed for both (a) and (b). When we have the initial speed, we can use this value to write the trajectory equation.
Solution
(a)
(b)
(c)
(d) Using a graphing utility, we can compare the two trajectories, which are shown in
[link] .
Significance
The initial speed for the shot at
is greater than the initial speed of the shot at
Note from
[link] that two projectiles launched at the same speed but at different angles have the same range if the launch angles add to
The launch angles in this example add to give a number greater than
Thus, the shot at
has to have a greater launch speed to reach 90 m, otherwise it would land at a shorter distance.
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miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
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Samuel
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