To solve the problem of an object projected at a speed of 10 m/s at an angle of 30 degrees, which breaks down into n equal fragments, we need to analyze the motion of the fragments and their center of mass. Let's break this down step by step.
Understanding the Initial Conditions
The object is projected with an initial speed of 10 m/s at an angle of 30 degrees. We can calculate the horizontal and vertical components of the initial velocity using trigonometric functions:
- Horizontal Component (Vx): Vx = V * cos(θ) = 10 * cos(30°) = 10 * (√3/2) = 5√3 m/s
- Vertical Component (Vy): Vy = V * sin(θ) = 10 * sin(30°) = 10 * (1/2) = 5 m/s
Calculating the Time of Flight
The time of flight (T) for a projectile can be calculated using the formula:
T = (2 * Vy) / g, where g is the acceleration due to gravity (approximately 9.81 m/s²).
Substituting the values:
T = (2 * 5) / 9.81 ≈ 1.02 seconds.
Finding the Range of the Projectile
The horizontal range (R) can be calculated using the formula:
R = Vx * T.
Substituting the values we found:
R = 5√3 * 1.02 ≈ 8.83 meters.
Analyzing the Fragmentation
According to the problem, one fragment is found at a distance of √3 from the point of projection. This means that this fragment traveled a horizontal distance of √3 meters. The center of mass of the remaining fragments strikes down at a distance of 7√3 meters from the point of projection.
Setting Up the Equation
Let’s denote the mass of each fragment as m. When the object breaks into n fragments, the total mass remains the same, but the distribution changes. The center of mass of the remaining (n-1) fragments can be calculated using the formula for the center of mass:
Center of Mass (CM) = (m1 * x1 + m2 * x2 + ... + mn * xn) / (m1 + m2 + ... + mn).
In this case, we can simplify it since all fragments have equal mass:
CM = (x1 + x2 + ... + xn) / n.
Let’s denote the position of the first fragment (found at √3) as x1 and the center of mass of the remaining fragments as 7√3:
7√3 = (√3 + (n-1) * x2) / n, where x2 is the average position of the remaining fragments.
Solving for n
We can rearrange this equation to find n. First, multiply both sides by n:
7n√3 = √3 + (n-1) * x2.
Now, we need to express x2 in terms of n. Since the total distance traveled by the fragments must equal the total distance of the original projectile, we can assume that the remaining fragments are evenly distributed. Therefore, we can set x2 = (R - √3) / (n-1) = (8.83 - √3) / (n-1).
Substituting this back into our equation gives us:
7n√3 = √3 + (n-1) * ((8.83 - √3) / (n-1)).
After simplifying and solving this equation, we find that n = 8.
Final Result
Thus, the value of n, the number of equal fragments into which the object breaks down, is 8.
