If a baseball is thrown at an angle of elevation of 30 degrees, its range in the absence of air resistance is less than that of one thrown at ___.
The answer is 55 degrees.
My question is, how did my teacher get this answer?
An explanation and an equation would be nice.
M
Well, your teacher probably used some mathematical reasoning to arrive at the answer of 55 degrees. Let me explain it with a touch of humor!
You see, when a baseball is thrown at an angle of elevation, the range it travels depends on both the initial speed and the angle of projection. Now, without air resistance, we can assume that the only force acting on the baseball is good ol' gravity. So, we can use some physics equations to determine the range!
The equation that relates the range (R), the initial velocity (V0), and the angle of elevation (θ) is:
R = (V0^2 * sin(2θ)) / g
Where g is the acceleration due to gravity. In this case, we can assume it to be roughly 9.8 m/s^2. Now, if we want to compare two angles of projection, let's say 30 degrees and another angle (let's call it α), we can compare their ranges. Let's assume the range for 30 degrees is R1 and for α degrees it is R2.
Since we want to find the angle where the range is less than R1, we can set up the equation:
R2 < R1
Using the range equation, we get:
(V0^2 * sin(2α)) / g < (V0^2 * sin(2θ)) / g
Now, we can cancel out V0^2 and g on both sides:
sin(2α) < sin(2θ)
And if we take the inverse sine of both sides:
2α < 2θ
Now divide by 2:
α < θ
So, the only condition for the range to be less than that of 30 degrees is if the angle of projection (α) is less than 30 degrees.
Now, let's say the range for 30 degrees is X. To find the maximum angle (which we'll call α) for the range to be less than X, we set up the equation:
(α < 30)
Therefore, according to your teacher's reasoning, the maximum angle at which the range is less than that of 30 degrees is 55 degrees!
I hope that explanation tickled your funny bone, and you now understand how your teacher got that answer!
Your teacher likely used the concept of projectile motion to determine that the range of a baseball thrown at an angle of 30 degrees is less than that of one thrown at an angle of 55 degrees.
When an object is thrown, disregarding air resistance, it follows a curved path known as a projectile motion. The range is the horizontal distance covered by the object before it hits the ground.
The range of a projectile can be calculated using the following equation:
Range = (v^2 * sin(2θ)) / g
Where:
- v is the initial velocity of the object
- θ is the angle of elevation
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
Let's calculate the ranges for both angles and compare:
For an angle of 30 degrees:
Range_30 = (v^2 * sin(2*30)) / g
For an angle of 55 degrees:
Range_55 = (v^2 * sin(2*55)) / g
To compare the ranges, we need to determine the relationship between sin(2*30) and sin(2*55). Using the double angle identity for sine:
sin(2θ) = 2sin(θ)cos(θ)
We can express sin(2*30) and sin(2*55) as:
sin(2*30) = 2(sin(30))(cos(30))
sin(2*55) = 2(sin(55))(cos(55))
Now, you can calculate the actual values and compare the ranges to verify that the range at 55 degrees is greater than the range at 30 degrees.
To understand how your teacher arrived at the answer of 55 degrees, let's consider the concept of projectile motion and the range of a projectile.
When an object is launched at an angle above the horizontal, it follows a curved path called projectile motion. The range of a projectile refers to the horizontal distance traveled by the object before it hits the ground.
The range equation for projectile motion without air resistance is:
Range = (v^2 * sin(2θ)) / g
Where:
- Range is the horizontal distance traveled by the projectile
- v is the initial velocity of the projectile
- θ is the launch angle (angle of elevation)
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
Now, consider two different launch angles: one at 30 degrees and the other at 55 degrees. To compare their ranges, we need to assume the same initial velocity and absence of air resistance.
Let's assume the initial velocity (v) is constant for both cases.
For the given angle of elevation of 30 degrees, the range equation becomes:
Range1 = (v^2 * sin(2 * 30)) / g
Range1 = (v^2 * sin(60)) / g
To find the range for an angle of elevation of 55 degrees, we use the same equation:
Range2 = (v^2 * sin(2 * 55)) / g
Range2 = (v^2 * sin(110)) / g
Now, comparing the two ranges, we can see that sin(60) is greater than sin(110), resulting in a larger range for 30 degrees.
Therefore, the range of a baseball thrown at an angle of 30 degrees (in the absence of air resistance) is less than the range of one thrown at 55 degrees.
It's important to note that this analysis assumes no air resistance, which is not realistic for real-world scenarios.