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a ball thrown in air follows a path given by y=x/√3-3gx^2/8 met. where xaxis is taken along the horizontal and yaxis long the vertical . the maximum displacement of the ball along x direction for which displacement along y is zero
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a ball thrown in air follows a path given by y=x/√3-3gx^2/8 met. where...
Solution:
To find the maximum displacement of the ball along the x-direction when the displacement along the y-direction is zero, we need to set the equation equal to zero and solve for x. Let's break down the problem step by step.
Step 1: Set the equation equal to zero.
y = x/√3 - 3gx^2/8
Since we want to find the maximum displacement of x when y is zero, we can set y equal to zero and solve for x.
0 = x/√3 - 3gx^2/8
Step 2: Simplify the equation.
To simplify the equation, we can multiply the entire equation by 8 to eliminate the fractions.
0 = 8(x/√3) - 8(3gx^2/8)
0 = 8x/√3 - 3gx^2
Step 3: Rearrange the equation.
To make the equation easier to work with, let's rearrange it.
3gx^2 = 8x/√3
Step 4: Square both sides of the equation.
To eliminate the square root, we can square both sides of the equation.
(3gx^2)^2 = (8x/√3)^2
9g^2x^4 = 64x^2/3
Step 5: Simplify the equation.
To simplify the equation, we can multiply both sides by 3 to eliminate the fraction.
27g^2x^4 = 64x^2
Step 6: Solve for x.
To solve for x, we can divide both sides of the equation by x^2.
27g^2x^2 = 64
x^2 = 64 / 27g^2
x = ±√(64 / 27g^2)
Step 7: Determine the maximum displacement.
Since we are looking for the maximum displacement along the x-direction, we only consider the positive value of x.
Therefore, the maximum displacement of the ball along the x-direction for which the displacement along the y-direction is zero is given by:
x = √(64 / 27g^2)
Conclusion:
The maximum displacement of the ball along the x-direction for which the displacement along the y-direction is zero is √(64 / 27g^2).
To find the maximum displacement of the ball along the x-direction when the displacement along the y-direction is zero, we need to set the equation equal to zero and solve for x. Let's break down the problem step by step.
Step 1: Set the equation equal to zero.
y = x/√3 - 3gx^2/8
Since we want to find the maximum displacement of x when y is zero, we can set y equal to zero and solve for x.
0 = x/√3 - 3gx^2/8
Step 2: Simplify the equation.
To simplify the equation, we can multiply the entire equation by 8 to eliminate the fractions.
0 = 8(x/√3) - 8(3gx^2/8)
0 = 8x/√3 - 3gx^2
Step 3: Rearrange the equation.
To make the equation easier to work with, let's rearrange it.
3gx^2 = 8x/√3
Step 4: Square both sides of the equation.
To eliminate the square root, we can square both sides of the equation.
(3gx^2)^2 = (8x/√3)^2
9g^2x^4 = 64x^2/3
Step 5: Simplify the equation.
To simplify the equation, we can multiply both sides by 3 to eliminate the fraction.
27g^2x^4 = 64x^2
Step 6: Solve for x.
To solve for x, we can divide both sides of the equation by x^2.
27g^2x^2 = 64
x^2 = 64 / 27g^2
x = ±√(64 / 27g^2)
Step 7: Determine the maximum displacement.
Since we are looking for the maximum displacement along the x-direction, we only consider the positive value of x.
Therefore, the maximum displacement of the ball along the x-direction for which the displacement along the y-direction is zero is given by:
x = √(64 / 27g^2)
Conclusion:
The maximum displacement of the ball along the x-direction for which the displacement along the y-direction is zero is √(64 / 27g^2).
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