To solve this problem, we need to break it down into several steps. First, we will analyze the projectile's motion before the explosion, then determine the conditions right after the explosion, and finally calculate the distance between the two components one second after the explosion.
Step 1: Analyze the Initial Projectile Motion
The projectile is launched with an initial speed of 24 m/s at an angle of 60° from the horizontal. We can find the initial horizontal and vertical components of the velocity using trigonometric functions:
- Horizontal velocity (Vx): Vx = V * cos(θ) = 24 * cos(60°) = 24 * 0.5 = 12 m/s
- Vertical velocity (Vy): Vy = V * sin(θ) = 24 * sin(60°) = 24 * (√3/2) ≈ 20.78 m/s
Step 2: Determine the Time to Reach the Highest Point
The time taken to reach the highest point can be calculated using the formula:
t = Vy / g
where g is the acceleration due to gravity (approximately 9.81 m/s²).
Substituting the values:
t = 20.78 / 9.81 ≈ 2.11 seconds
Step 3: Calculate the Maximum Height
To find the maximum height (H) reached by the projectile, we can use the formula:
H = Vy * t - (1/2) * g * t²
Substituting the values we have:
H = 20.78 * 2.11 - 0.5 * 9.81 * (2.11)²
H ≈ 43.73 - 21.74 ≈ 21.99 meters
Step 4: Conditions After the Explosion
At the highest point, the projectile explodes into two equal masses. One component travels vertically upward with a speed of 7/2 m/s (which is 3.5 m/s). The other component will continue moving horizontally with the same horizontal velocity (12 m/s) since there are no horizontal forces acting on it.
Step 5: Calculate the Positions After One Second
Now, we will find the positions of both components one second after the explosion:
Vertical Component
The vertical component moves upward with an initial speed of 3.5 m/s. The vertical position (y) after one second can be calculated as:
y = H + (initial vertical speed * time) - (1/2) * g * time²
y = 21.99 + (3.5 * 1) - (0.5 * 9.81 * 1²)
y ≈ 21.99 + 3.5 - 4.905 ≈ 20.585 meters
Horizontal Component
The horizontal component continues to move horizontally at 12 m/s. The horizontal position (x) after one second is:
x = initial horizontal speed * time
x = 12 * 1 = 12 meters
Step 6: Calculate the Distance Between the Two Components
Now we can find the distance between the two components using the Pythagorean theorem. The horizontal distance (x) is 12 meters, and the vertical distance (y) is the difference in their heights:
Vertical distance = 21.99 - 20.585 = 1.405 meters
Using the Pythagorean theorem:
Distance = √(x² + y²) = √(12² + 1.405²)
Distance = √(144 + 1.973) ≈ √145.973 ≈ 12.08 meters
Thus, one second after the explosion, the distance between the two components is approximately 12.08 meters.
