91Ó°ÊÓ

Kinetic energy is the energy possessed by an object due to its motion. It's a crucial concept when discussing motion-related problems, especially those involving bullets or fast-moving objects. The kinetic energy (\( KE \)) of an object can be calculated using the formula: \[ KE = 0.5 \cdot m \cdot v^2 \]where \( m \) is the mass in kilograms, and \( v \) is the velocity in meters per second.

• In this problem, the mass of the bullet is \( 7.80 \text{ grams} = 0.0078 \text{ kg}. \)
• The speed of the bullet is \( 575 \text{ m/s}. \)

Inserting these known values into the formula gives the initial kinetic energy. As the bullet penetrates into the tree, this energy is gradually dissipated until it reaches zero, indicating the bullet has come to a stop.

Frictional force acts opposite to the direction of motion, and it's the key factor responsible for stopping the bullet in this exercise. When an object like a bullet encounters a different material, in this case, the trunk of a tree, it experiences a resistant force.

• The work done by friction, indicated by \( W \), is equivalent to the change in kinetic energy as the bullet stops.
• Since the bullet ceases moving, the final kinetic energy is zero, and the work done can be expressed as a loss of initial kinetic energy: \[ W = - KE_i. \]

The average frictional force \( F \) can be calculated using the formula \[ F = \frac{-W}{d} \], where \( d \) is the penetration depth of \( 0.055 \text{ meters}. \)This force, being constant in this scenario, directly influences the deceleration needed to halt the motion.

Deceleration is the rate at which an object slows down. It's essentially the opposite of acceleration and is a direct consequence of the frictional force acting on the bullet. The deceleration (\( a \)) can be calculated using Newton's formula for force:\[ F = m \cdot a \]. Here, knowing the force (\( F \)) from the previous section and the mass (\( m \)) of the bullet, the deceleration can be determined using:

• \[ a = \frac{F}{m}. \]

This negative acceleration value indicates how the velocity of the bullet decreases over the distance traveled until it reaches a complete stop.

Newton's second law defines the relationship between the motion of an object and the forces acting on it. It is expressed as \( F = m \cdot a \), where force (\( F \)) is the product of mass (\( m \)) and acceleration (\( a \)). When the bullet hits the tree, the significant frictional force works against its motion, causing it to decelerate.The formula helps us calculate the deceleration in the problem. Additionally, by rearranging the equation \[ v_f = v_i + a \cdot t \], with \( v_f = 0 \), we can solve for the time (\( t \)) required for the bullet to come to rest:

• \[ t = \frac{-v_i}{a} \]

This time represents how long it takes for the bullet to lose its initial velocity under the constant deceleration caused by the friction in the tree, giving insight into both motion and force dynamics.