A 47.0 g golf ball is driven from the tee with an initial speed of 48.0 m/s and rises to a height of 23.2 m.
(a) Neglect air resistance and determine the kinetic energy of the ball at its highest point.
(b) What is its speed when it is 8.0 m below its highest point?
Y^2 = Yo^2 + 2g*h = 0
Yo^2 - 19.6*23.2 = 0
Yo^2 = 454.72
Yo = 21.3 m/s = Ver. component of initial velocity.
Yo = Vo*sin A = 21.3 m/s
48*sin A = 21.3
sin A = 0.44,425
A = 26.4o = Angle at which golf was hit.
Vo = 48m/s[26.4o]
Xo = 48*Cos26.4 = 43.0 m/s.
V = Xo + Yi = 43 + 0i = 43 m/s at max. ht.
a. KE = 0.5M*V^2 = 0.5*0.047*43^2 = 43.5 J.
b. Y^2 = Yo^2 + 2g*h = 0 + 19.6(23.2-8) = 298
Y = 17.3 m/s = Ver. component.
V = sqrt (Xo^2 + Y^2)
Xo = 43 m/s
Y = 17.3 m/s
Solve for V.
v=51.51m/s
To solve this problem, we need to analyze the motion of the golf ball using the principles of energy conservation. We can break down the problem into two parts:
(a) Determining the kinetic energy at the highest point:
At the highest point, the golf ball has reached its maximum height, and we can assume that all of its initial kinetic energy has been converted into potential energy. Since there is no information given about the presence of any external forces, we can conclude that the total mechanical energy (kinetic energy + potential energy) of the ball is conserved.
The initial kinetic energy (KE_init) of the ball can be calculated using the formula:
KE_init = 1/2 * m * v^2
where m is the mass of the ball (47.0 g = 0.047 kg) and v is its initial velocity (48.0 m/s).
KE_init = 1/2 * 0.047 kg * (48.0 m/s)^2
KE_init = 54.144 J
Since the ball reaches its highest point, all of its initial kinetic energy is converted into potential energy (PE_max).
Therefore, the kinetic energy of the ball at its highest point is 54.144 J.
(b) Determining the speed when the ball is 8.0 m below its highest point:
To solve this part, we can use the principle of conservation of mechanical energy again. At a height 8.0 m below the highest point, the mechanical energy of the ball will be equal to the sum of its kinetic and potential energy at that point.
The mechanical energy (ME) of the ball can be calculated as follows:
ME = KE + PE
where KE is the kinetic energy and PE is the potential energy.
At this point, the potential energy (PE) can be calculated using the formula:
PE = m * g * h
where m is the mass of the ball (0.047 kg), g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height (8.0 m).
PE = 0.047 kg * 9.8 m/s^2 * 8.0 m
PE = 3.6864 J
Since the total mechanical energy is conserved, we can write the following equation:
ME = KE + PE
At the highest point, the kinetic energy was entirely converted to potential energy, so at this point, the potential energy is equal to the total mechanical energy.
ME = PE_max = 54.144 J
ME at 8 m below highest point = KE + PE = 54.144 J
KE = ME - PE
KE = 54.144 J - 3.6864 J
KE ≈ 50.4576 J
To determine the speed, we can use the formula for kinetic energy and the mass of the golf ball:
KE = 1/2 * m * v^2
Rearranging the equation to solve for velocity (v):
v = √(2 * KE / m)
v = √(2 * 50.4576 J / 0.047 kg)
Calculating the result:
v ≈ √(2149.68)
v ≈ 46.35 m/s
Therefore, the speed of the ball when it is 8.0 m below its highest point is approximately 46.35 m/s.