1 A hot-air balloon is rising upwards at a constant velocity of 5m.s When the balloon is 100m above the ground, a sandbag is dropped from it.
1.1 what is the velocity of the sandbag when its displacement is zero.
1.2 Determine the maximum height X, above the ground reached by the sandbag after it is released from the balloon.
1.1 To find the velocity of the sandbag when its displacement is zero, we can use the principle of conservation of mechanical energy. When the sandbag is dropped, it starts with an initial velocity of 0 and only experiences gravitational potential energy. As it falls, this potential energy is converted into kinetic energy. When the displacement is zero, its potential energy is fully converted into kinetic energy.
To calculate the velocity, we'll use the equation for gravitational potential energy:
Potential Energy (PE) = m * g * h
Where:
m is the mass of the sandbag,
g is the acceleration due to gravity (approximately 9.8 m/s²),
h is the height above ground.
Since the initial velocity is zero, initially all the potential energy is converted to kinetic energy, so we can set PE equal to the kinetic energy:
Kinetic Energy (KE) = (1/2) * m * v²
Setting the two equations equal:
m * g * h = (1/2) * m * v²
Canceling the mass m on both sides:
g * h = (1/2) * v²
Simplifying the equation:
v² = 2 * g * h
Taking the square root of both sides:
v = sqrt(2 * g * h)
Plugging in the values:
g = 9.8 m/s²
h = 100m
v = sqrt(2 * 9.8 * 100)
v = sqrt(1960)
v ≈ 44.27 m/s
Therefore, the velocity of the sandbag when its displacement is zero is approximately 44.27 m/s.
1.2 To determine the maximum height (X) reached by the sandbag after it is released, we can use the following equation:
Final Velocity (vf)² = Initial Velocity (vi)² + 2 * acceleration (a) * displacement (d)
Initially, the sandbag has a velocity of approximately 44.27 m/s (as derived in 1.1). Since the final velocity is 0 when the displacement is at its maximum, we can rewrite the equation as:
0 = 44.27² + 2 * (-9.8) * X
Simplifying the equation:
0 = 1960 + (-19.6) * X
Rearranging the equation to solve for X:
19.6 * X = 1960
Dividing both sides by 19.6:
X = 1960 / 19.6
X ≈ 100 m
Therefore, the maximum height reached by the sandbag after it is released from the balloon is approximately 100m above the ground.
1.1 In order to find the velocity of the sandbag when its displacement is zero, we need to consider the motion of the sandbag after it is dropped from the balloon.
When the sandbag is dropped, it will experience a downward acceleration due to gravity. However, since the balloon is rising upwards at a constant velocity, the sandbag will also have an initial upward velocity equal to the velocity of the balloon.
Since the sandbag is initially moving upwards with the same velocity as the balloon, and later reaches a point where its displacement is zero, this means that the sandbag will eventually come to rest and start moving downwards. At this point, its velocity will be zero.
Therefore, the velocity of the sandbag when its displacement is zero is 0 m/s.
1.2 To determine the maximum height reached by the sandbag after it is released from the balloon, we can use the equations of motion.
The initial velocity of the sandbag is equal to the velocity of the balloon, which is 5 m/s (given).
The acceleration of the sandbag is the acceleration due to gravity, which is approximately 9.8 m/s^2.
The initial displacement of the sandbag is 100 m (given).
Using the equation of motion:
v^2 = u^2 + 2as
where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the displacement, we can solve for the final velocity of the sandbag when it reaches maximum height:
0 = (5)^2 + 2(-9.8)s
Simplifying the equation:
0 = 25 - 19.6s
19.6s = 25
s = 25 / 19.6
s ≈ 1.275 m
Therefore, the maximum height reached by the sandbag after it is released from the balloon is approximately 1.275 m above the ground.