A coin is dropped from a hot-air balloon that is 300 m above the ground and rising at 10.0 m / s

A coin is dropped from a hot-air balloon that is 300 m above...

Key Concepts

Initial Conditions in Vertical Motion

In any analysis of vertical motion, it is crucial to consider the initial conditions, such as the initial height and initial velocity. In this problem, the coin starts with a positive upward velocity because it is released from a rising balloon, and its initial height is above the ground. These conditions determine how the coin’s trajectory develops over time.

Kinematic Equations

Kinematic equations provide the mathematical relationships between displacement, initial velocity, final velocity, acceleration, and time in uniformly accelerated motion. These equations are fundamental tools for solving problems that involve calculating positions, velocities, maximum heights, and the duration of flight in projectile motion scenarios.

Uniformly Accelerated Motion

This refers to any motion where an object experiences a constant acceleration, such as the acceleration due to gravity near the Earth's surface. In problems like this, the object’s motion can be analyzed using standard kinematic equations, which assume that the acceleration (in this case, gravity) remains constant throughout the motion.

Projectile Motion

Projectile motion involves the movement of objects that are launched into the air and then move under the influence of gravity. Understanding projectile motion is essential to analyze the trajectory, maximum height, time to reach the ground, and the effects of initial velocity when an object is released or thrown from a moving platform.

Transcript

00:02 First, when a coin is dropped from a hot air balloon that is rising upward with the speed of 10 m per second, and then the coin acquires an initial motion of 10 meter per second.

00:15 Consider the upward direction as the positive and downward is negative.

00:20 The motion of coin is under gravity, therefore its acceleration due to gravity is minus 9 .8 meter per second square.

00:29 Also, when the coin reaches the maximum, height then its velocity becomes zero then consider the motion of the coin when it is thrown from the hot air balloon to the maximum height the maximum height that the coin reaches is calculated by the kinematic equation of motion for velocity and proportion that is b square is equal to v0 square plus twice a y minus y 0 y minus y0 is the displacement so we have v square is 0 and v not is 10 meter per second square plus twice a is minus 9 .8 and y minus y 0 that is 10 meter square per second square is equal to 19 .6 meter per second square is equal to 19 .6 meter per second.

01:36 Minus y 0 y minus y 0 is 100 meters square per square second square upon 19 .6 meter per second square that is 5 .1 meter therefore the maximum height from the ground is from the ground is 300 meter plus 5 .1 one meter that is 305 .1 meter from the ground now take the initial position of the coin one it is thrown as a reference then its initial position is zero and if initial position zero final position for second the position of the coin after four second is calculated by the kinematic equation of motion for position and time that is y minus y not is equal to v0 t i wrote t plus half a t square we have y minus y y not is zero not ten meter per second t is four second plus half a is minus 9 .8 for second square this y is minus 38 .4 meter this is the height of the coin from the hot air balloon so it's height from the ground after four second is 300 meters minus 38 .4 meter is 261 .6 meter.

04:38 Now the final velocity of the coin after 4 second is calculated by the dynamic equation of motion for velocity and acceleration.

04:52 This is v is equal to v0 plus a t 10 meter plus minus 9 .8.

05:04 Meter per second square t for second that is minus 29 .2 meter per hence the position of the coin after four second is to 61 .6 and its velocity at this point is minus 29 point now consider the motion of the coin into different k first condition consider the motion of the coin from the hot tub alone to the maximum height and the second condition condition can consider the motion from the maximum height to the ground.

05:48 Consider the first case in which the coin is in motion from the hot air balloon to the maximum height.

05:54 The time for the coin to reach the maximum height is calculated by the kinematic equation of motion for velocity and acceleration, that is v v0 plus a, t...

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