Askiitians Tutor Team
Last Activity: 4 Months ago
To solve this problem, we need to apply the principles of conservation of momentum and projectile motion. The boy and the platform will move in the opposite direction to the stone when it is thrown, due to the conservation of momentum. Let's break this down step by step.
Understanding the System
We have a boy with a mass of 60 kg standing on a platform that has a mass of 40 kg. Together, they form a system with a total mass of 100 kg. When the boy throws a stone of mass 1 kg at an angle of 45 degrees with an initial velocity of 10 m/s, the stone will follow a projectile motion path.
Calculating the Stone's Velocity Components
First, we need to find the horizontal and vertical components of the stone's velocity. Since the stone is thrown at an angle of 45 degrees, we can use trigonometric functions:
- Horizontal component (Vx) = V * cos(θ) = 10 m/s * cos(45°) = 10 m/s * (1/√2) ≈ 7.07 m/s
- Vertical component (Vy) = V * sin(θ) = 10 m/s * sin(45°) = 10 m/s * (1/√2) ≈ 7.07 m/s
Time of Flight
Next, we need to determine how long the stone will be in the air. The time of flight for a projectile can be calculated using the formula:
Time of flight (T) = (2 * Vy) / g
Substituting the values:
T = (2 * 7.07 m/s) / 10 m/s = 1.414 seconds
Momentum Conservation
According to the law of conservation of momentum, the total momentum before the stone is thrown must equal the total momentum after it is thrown. Initially, the system (boy + platform + stone) is at rest, so the initial momentum is 0.
After the stone is thrown, the momentum of the stone and the momentum of the boy and platform must balance out:
Let the velocity of the platform and the boy after the stone is thrown be Vp.
Momentum of the stone = mass of stone * velocity of stone = 1 kg * 7.07 m/s = 7.07 kg·m/s
Momentum of the boy and platform = (mass of boy + mass of platform) * Vp = 100 kg * Vp
Setting the two momenta equal gives us:
0 = 7.07 kg·m/s - 100 kg * Vp
Solving for Vp:
Vp = 7.07 kg·m/s / 100 kg = 0.0707 m/s
Calculating the Displacement
Now that we have the velocity of the platform and the boy, we can find the displacement of the platform during the time the stone is in the air. The displacement (d) can be calculated using the formula:
d = Vp * T
Substituting the values we found:
d = 0.0707 m/s * 1.414 s ≈ 0.1 m
Final Result
The displacement of the platform (with the boy) on the horizontal surface when the stone lands on the ground is approximately 0.1 meters, or 10 centimeters. This means that as the boy threw the stone, the platform moved slightly backward due to the conservation of momentum.
