A PROJECTILE HAS SAME RANGE 'R' WHEN MAXIMUM HEIGHT ATTAINED BY IT IS EITHER 'H1' OR 'H2'.FIND THE RELATION BETWEEN,'R','H1','H2'.
The range of a projectile fired at angle A with velocity V is
R = 2(V^2/g)*sinA*cosA
Rmax = (V^2/g) (obtained when A = 45 degrees)
At a given V, you get the same range R when the two angles A1 and A2 are complementary. For example,
R(50 degrees) = R(40 degrees)
Let A1>A2
As for the heights,
H1 = V^2 sin^2A1/(2g)
H2 = V^2 sin^2A2/(2g)
= V^2(cos^2A1)/(2g)
H1^2 + H2^2 = V^4/(4g^2) =
(1/4)*Rmax^2
Where Rmax is the maximum possible range, which is obtained when A = 45 degrees
Correct
To find the relation between the range (R), maximum height attained (H1), and another maximum height attained (H2) by a projectile with the same range, we can use the following equation:
R = [(v^2) * sin (2θ)] / g
Where:
R is the range of the projectile,
v is the initial velocity of the projectile,
θ is the launch angle of the projectile,
g is the acceleration due to gravity.
To determine the relation between R, H1, and H2, let's consider the two cases separately:
Case 1: Maximum Height H1
In this case, the maximum height attained by the projectile is H1. Using the kinematic equation for vertical motion, we can determine the equation for H1:
H1 = (v^2 * sin^2θ) / (2g)
Case 2: Maximum Height H2
In this case, the maximum height attained by the projectile is H2. Using the same kinematic equation for vertical motion, we can determine the equation for H2:
H2 = (v^2 * sin^2θ) / (2g)
Since we are given that the range (R) is the same for both cases, we can equate the expressions for R obtained from both cases:
[(v^2) * sin (2θ)] / g = [(v^2 * sin^2θ) / (2g)]
Simplifying this equation will allow us to find the relation between R, H1, and H2:
2 * sin (2θ) = sin^2θ
Expanding and simplifying further:
2 * 2 * sin (θ) * cos (θ) = sin^2θ
4 * sin (θ) * cos (θ) = sin^2θ
2 * sin (θ) * cos (θ) = (1 - cos^2θ)
Rearranging the equation:
2 * sin (θ) * cos (θ) + cos^2θ = 1
Using the identity sin^2θ + cos^2θ = 1:
2 * sin (θ) * cos (θ) + (1 - sin^2θ) = 1
Simplifying further:
2 * sin (θ) * cos (θ) - sin^2θ + 1 = 1
Now, combining like terms:
- sin^2θ + 2 * sin (θ) * cos (θ) + 1 = 1
Rearranging again:
- sin^2θ + 2 * sin (θ) * cos (θ) = 0
Factoring:
sin (θ) * (2 * cos (θ) - sin (θ)) = 0
Since sin(θ) cannot be zero:
2 * cos (θ) - sin (θ) = 0
Simplifying:
2 * cos (θ) = sin (θ)
Dividing both sides by cos(θ):
2 = tan(θ)
Therefore, the relation between R, H1, and H2 when the projectile has the same range is:
H1 = H2
In other words, the maximum heights attained in both cases are the same.
To find the relationship between the range (R) of a projectile and the maximum heights attained (H1 and H2), we can make use of the equations of motion for a projectile.
Let's break down the problem step by step:
1. Range (R): The range of a projectile is the horizontal distance covered by it from the point of projection to the point where it hits the ground. The formula for range is given by R = (v^2 * sin(2θ)) / g, where v is the initial velocity of the projectile, θ is the launch angle, and g is the acceleration due to gravity.
2. Maximum height (H): The maximum height attained by a projectile is the vertical distance it reaches above the point of projection. The formula for maximum height is given by H = (v^2 * sin^2(θ)) / (2g), where v, θ, and g have the same meanings as before.
Given that the range is the same (R) for both H1 and H2, we can set up the following equation:
R = (v^2 * sin(2θ)) / g = (v^2 * sin^2(θ)) / (2g)
We can simplify this equation to find the relationship between R, H1, and H2:
2 * sin(2θ) * g = sin^2(θ)
Now, we just need to solve this equation to get the desired relationship.
Keep in mind that solving this equation may require applying trigonometric identities to simplify it further.
Alternatively, if you have specific values for R, H1, or H2, you can substitute those values into the equation and solve it accordingly, allowing you to find the relationship directly.
Remember, the relationship between R, H1, and H2 may vary depending on the values of the initial velocity (v) and launch angle (θ).