To determine the difference in the speed of the two balls when they hit the ground, we can use the principle of conservation of energy. This principle states that the total mechanical energy of a system remains constant if no external forces are acting on it.
Let's break down the problem step by step:
Step 1: Calculate the initial potential energy of each ball.
The potential energy of an object near the surface of the Earth can be expressed as: PE = mgh, where m is the mass of the object, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height.
For each ball, calculate the initial potential energy:
PE1 = mgh = 0.550 kg * 9.8 m/s^2 * 2.30 m
PE2 = mgh = 0.550 kg * 9.8 m/s^2 * 2.30 m
Step 2: Calculate the final kinetic energy of each ball.
Since both balls are dropped, without any initial velocity, their initial kinetic energy is zero. When they hit the ground, all the potential energy is converted into kinetic energy.
The final kinetic energy can be calculated using KE = 0.5mv^2, where m is the mass of the object and v is the final velocity.
For each ball, calculate the final kinetic energy:
KE1 = 0.5 * 0.550 kg * v1^2
KE2 = 0.5 * 0.550 kg * v2^2
Step 3: Calculate the final velocity of each ball using the relation between kinetic energy and potential energy.
Since the total mechanical energy is conserved, we can equate the initial potential energy to the final kinetic energy:
PE1 + PE2 = KE1 + KE2
Substituting the values obtained from steps 1 and 2, the equation becomes:
0.550 kg * 9.8 m/s^2 * 2.30 m + 0.550 kg * 9.8 m/s^2 * 2.30 m = 0.5 * 0.550 kg * v1^2 + 0.5 * 0.550 kg * v2^2
Step 4: Solve for the difference in the squares of the velocities.
Rearrange the equation to isolate the difference in the squares of the velocities:
v1^2 - v2^2 = (2 * (0.550 kg * 9.8 m/s^2 * 2.30 m)) / 0.550 kg
Step 5: Take the square root of both sides of the equation to solve for the difference in velocities.
sqrt(v1^2 - v2^2) = sqrt((2 * (0.550 kg * 9.8 m/s^2 * 2.30 m)) / 0.550 kg)
Calculating this equation will give you the difference in the speeds of the two balls when they hit the ground.