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Kinetic energy is a form of energy that an object possesses due to its motion. It depends on two key factors: the mass of the object and its speed. The formula to calculate kinetic energy (KE) is given by \(KE = \frac{1}{2}mv^2\), where \(m\)is the mass in kilograms, and \(v\)is the velocity in meters per second. In our example, the projectile has an initial velocity of 18.0 m/sand a mass of 0.750 kg. When it was launched, it had a kinetic energy calculated as follows:

• \( KE = \frac{1}{2} (0.750 \, \text{kg})(18.0 \, \text{m/s})^2\)
• This gives a kinetic energy of \(121.5 \, \text{Joules}\)just as it leaves the ground.

As the projectile moves upwards, its kinetic energy transforms into potential energy. This transformation continues until the projectile reaches the peak of its trajectory, where its speed, and therefore its kinetic energy, is zero.

Potential energy is energy that is stored within an object due to its position or height. For an object lifted against gravity, its potential energy can be calculated using the formula:\(PE = mgh\),where \(m\)is the mass, \(g\) is the acceleration due to gravity, and \(h\) is the height above a reference point.In the projectile motion exercise, as the projectile rises, its kinetic energy is converted to potential energy. When the projectile reaches its highest point, its potential energy is at maximum:

• Without air resistance, using height 16.53 m,
• Potential Energy would be \(PE = 0.750 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 16.53 \, \text{m} \approx 121.5 \, \text{J}\).

This conversion happens because all the kinetic energy initially imparted is used to overcome gravitational potential energy as height increases.

Air resistance is a force that opposes the motion of an object through the air. It acts in the opposite direction to the object's movement and can significantly affect the motion of objects, especially at high speeds or with large surface areas. In our projectile problem, air resistance is the reason the projectile rises to only 11.8 m instead of reaching 16.53 m. Some key points about air resistance:

• It depends on factors such as the object's velocity, cross-sectional area, air density, and the drag coefficient.
• Higher speed means greater air resistance.
• In our case, air resistance causes a difference in how high the projectile goes compared to if it were in a vacuum.

Air resistance converts some of the projectile's kinetic energy to other forms of energy, like thermal energy, preventing the projectile from achieving its full potential height.

The work-energy principle links the concepts of work and energy. It states that the work done on an object equals the change in its kinetic energy. This principle can also be applied to potential energy changes, especially in projectile motion where energy conservation is a key concept.In our exercise, after calculating the potential energy difference:

• When no air resistance is present, potential energy at maximum height would be \(121.5 \, \text{J}\).
• With air resistance, potential energy is \(87.0 \, \text{J}\),using the height 11.8 m.

The difference of 34.5 Jis work done by air resistance:

• Using the equation \(W = F_{average} \times d\), where \(W\)is work done and \(d\)is distance over which the force is applied.
• This allows us to find the average force of air resistance using \(F_{average} = \frac{34.5 \text{J}}{11.8 \text{m}} \approx 2.92 \, \text{N}\).

Understanding this principle helps to connect work done against forces and energy transformations during movements like projectile motion.