A small rocket burns 0.0500 kg of fuel per second, ejecting it as a gas with a velocity relative to the rocket of magnitude 1600 m/s. Would the rocket operate in outer space where there is no atmosphere? If so, how would you steer it? Could you brake it?
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2. 1D Motion / Kinematics
Vertical Motion and Free Fall
Problem 56a
Textbook Question
A baseball is hit almost straight up into the air with a speed of 22 m/s. Estimate how high it goes.

1
Identify the known values: The initial velocity \( v_0 \) is 22 m/s, and the acceleration due to gravity \( g \) is approximately \( 9.8 \; \text{m/s}^2 \) (acting downward). At the highest point, the final velocity \( v \) will be 0 m/s because the baseball momentarily stops before falling back down.
Use the kinematic equation to relate the initial velocity, final velocity, acceleration, and displacement: \( v^2 = v_0^2 + 2a y \), where \( y \) is the maximum height, \( a \) is the acceleration (negative in this case), and \( v \) is the final velocity.
Substitute the known values into the equation: \( 0 = (22)^2 + 2(-9.8)y \). Simplify the equation to isolate \( y \), the maximum height.
Rearrange the equation to solve for \( y \): \( y = \frac{-(22)^2}{2(-9.8)} \). Simplify the numerator and denominator to express \( y \) in terms of the given values.
Perform the division to calculate the maximum height \( y \). This will give you the estimated height the baseball reaches before it starts descending.

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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Kinematic Equations
Kinematic equations describe the motion of objects under constant acceleration. In this scenario, the baseball is subject to gravitational acceleration as it moves upward. The relevant equation for maximum height is derived from the initial velocity and acceleration due to gravity, allowing us to calculate how high the baseball will rise before it starts to fall back down.
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Acceleration due to Gravity
Acceleration due to gravity is a constant value that represents the rate at which an object accelerates towards the Earth when in free fall. On Earth, this value is approximately 9.81 m/s². When the baseball is hit upwards, it decelerates at this rate until it reaches its peak height, where its velocity becomes zero before descending.
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Conservation of Energy
The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In the context of the baseball, the kinetic energy from the initial speed is converted into gravitational potential energy at the peak height. This relationship allows us to calculate the maximum height by equating the initial kinetic energy to the potential energy at the highest point.
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