91Ó°ÊÓ

The kinetic energy of an object tells us how much energy it has due to its motion. To calculate kinetic energy, we use the formula: \[ KE = \frac{1}{2} m v^2 \]where:

• \( m \) is the mass of the object (in kilograms).
• \( v \) is the velocity (in meters per second).

In the original exercise, we calculated the initial kinetic energy for a snowball.The snowball had a mass of \(1.50 \, \text{kg} \) and was moving at \(20.0 \, \text{m/s}\).By substituting these given values into the kinetic energy formula, the calculation was:\[ KE = \frac{1}{2} \times 1.50 \, \text{kg} \times (20.0 \, \text{m/s})^2 = 300 \, \text{J} \]Therefore, the initial kinetic energy of our snowball is \(300 \, \text{J}\).This amount of energy is what allows the snowball to ascend against gravity.The faster or heavier an object, the more kinetic energy it possesses.

Gravitational potential energy (GPE) comes into play when objects are moved against the force of gravity.It's important to understand how height affects the energy of an object in a gravitational field.GPE is calculated using the formula:\[ U = mgh \]where:

• \( m \) is the mass of the object.
• \( g \) stands for the acceleration due to gravity (\(9.81 \, \text{m/s}^2\) on Earth).
• \( h \) is the height (in meters) that the object is raised.

In the context of the exercise, we were interested in how GPE changed as the snowball rose to its highest point.From the solved step, we know the snowball reached a maximum height of \(6.38 \, \text{m}\).Using the formula above, the change in gravitational potential energy was:\[ \Delta U = 1.50 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 6.38 \, \text{m} \approx 93.92 \, \text{J} \]

When dealing with projectile motion, it's crucial to break the velocity into horizontal and vertical components.This helps in analyzing the motion in each direction separately.The vertical component of velocity \( v_y \) is found using:\[ v_y = v \cdot \sin(\theta) \]where:

• \( v \) is the overall initial velocity.
• \( \theta \) is the angle of projection with respect to the horizontal.

In our case, \( v = 20.0 \, \text{m/s} \) and \( \theta = 34.0^\circ \).The calculation becomes:\[ v_y = 20.0 \, \text{m/s} \times \sin(34.0^\circ) \approx 11.184 \, \text{m/s} \]This component dictates how high the snowball will travel before gravity halts its ascent.

The maximum height a projectile reaches is determined when the vertical component of its velocity becomes zero.This is the point where all its initial vertical kinetic energy has been converted into gravitational potential energy.To find this height, we use the equation:\[ v_y^2 = u_y^2 - 2gh \]where:

• \( v_y \) is the final vertical velocity (0 at maximum height).
• \( u_y \) is the initial vertical velocity.
• \( g \) is the acceleration due to gravity.
• \( h \) is the maximum height.

For the snowball, \( u_y = 11.184 \, \text{m/s} \), and \( g = 9.81 \, \text{m/s}^2 \).Rearranging and substituting, we calculated:\[ 0 = (11.184 \, \text{m/s})^2 - 2 \times 9.81 \, \text{m/s}^2 \times h \]\[ h = \frac{11.184^2}{2 \times 9.81} \approx 6.38 \, \text{m} \]This height reflects the snowball's peak ascent during its flight.