Final answer:
The speed of the ball, when it has fallen a distance of 1.75 m, is approximately 5.5 m/s.
Step-by-step explanation:
To find the speed of the ball when it has fallen a distance of 1.75 m, we can use the principle of conservation of energy.
First, let's find the potential energy (PE) of the ball when it has fallen a distance of 1.75 m. The potential energy is given by the equation PE = mgh, where m is the mass of the ball, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height.
PE = mgh = (5.00 kg)(9.8 m/s²)(1.75 m) = 85.75 J
Next, let's find the work done by friction. The work done by friction is equal to the force of friction multiplied by the distance the block moves. The force of friction can be calculated using the equation f = μmg, where μ is the coefficient of friction, m is the mass of the block, and g is the acceleration due to gravity.
f = μmg = (0.200)(3.00 kg)(9.8 m/s²) = 5.88 N
The work done by friction is given by the equation W = f * d, where W is the work done, f is the force of friction, and d is the distance.
W = f * d = (5.88 N)(1.75 m) = 10.29 J
Now, let's find the kinetic energy (KE) of the ball when it has fallen a distance of 1.75 m. The kinetic energy is given by the equation KE = 0.5 * mv^2, where m is the mass of the ball and v is the speed.
KE = 0.5 * mv²
Since the system starts from rest, the initial kinetic energy is 0. Therefore, the final kinetic energy is equal to the initial potential energy minus the work done by friction.
KE = PE - W = 85.75 J - 10.29 J = 75.46 J
Now we can solve for the speed of the ball using the equation for kinetic energy.
KE = 0.5 * mv²
75.46 J = 0.5 * (5.00 kg) * v²
Divide both sides by 0.5 * (5.00 kg) to solve for v²:
v² = (75.46 J) / (0.5 * (5.00 kg)) = 30.184 m²/s²
Finally, take the square root of both sides to find the speed of the ball:
v = √(30.184 m²/s²) ≈ 5.5 m/s
Therefore, the speed of the ball when it has fallen a distance of 1.75 m is approximately 5.5 m/s.