A 47.0 g golf ball is driven from the tee with an initial speed of 60.0 m/s and rises to a height of 24.4 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 7.0 m below its highest point?
To solve part (a) of the problem, we need to determine the kinetic energy of the golf ball at its highest point. Kinetic energy (KE) is given by the formula:
KE = 1/2 * m * v^2
Where:
- m is the mass of the golf ball
- v is the velocity (speed) of the golf ball
(a) To begin, we are given that the mass (m) of the golf ball is 47.0 g. However, we need to convert the mass to kilograms (kg) since the SI unit for mass is kg. We can do this by dividing the mass in grams by 1000:
m = 47.0 g / 1000 = 0.047 kg
Next, we are given the initial speed (v) of the golf ball is 60.0 m/s. We can directly use this value in the formula.
Substituting the values into the formula, we have:
KE = 1/2 * 0.047 kg * (60.0 m/s)^2
Calculating this expression, we get:
KE = 1/2 * 0.047 kg * 3600 m^2/s^2 = 81.72 J
Therefore, the kinetic energy of the golf ball at its highest point is 81.72 Joules.
(b) Now, let's solve part (b) of the problem to determine the speed of the golf ball when it is 7.0 m below its highest point.
To find the speed, we need to use the concept of conservation of energy. At the highest point, the sum of the gravitational potential energy (GPE) and the kinetic energy (KE) of the golf ball will be equal to the total energy (TE). Since there is no air resistance, the total energy remains constant throughout the motion.
The total energy (TE) is given by:
TE = GPE + KE
At the highest point, the golf ball has its maximum potential energy and zero kinetic energy. Since the height at the highest point is 24.4 m, its gravitational potential energy is:
GPE = m * g * h
Where:
- m is the mass of the golf ball
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
- h is the height (24.4 m)
Substituting the values into the formula, we have:
GPE = 0.047 kg * 9.8 m/s^2 * 24.4 m = 11.0036 J
Since the total energy (TE) is conserved, we can write the equation:
TE = GPE + KE
At the highest point, TE = GPE since KE is zero. So, we have:
11.0036 J = GPE + 0 J
11.0036 J = GPE
Now, when the golf ball is at a height of 7.0 m below its highest point, the gravitational potential energy will be:
GPE = m * g * h
Substituting the values, we have:
GPE = 0.047 kg * 9.8 m/s^2 * 7.0 m = 3.2116 J
Since the total energy (TE) is still conserved, we can write:
TE = GPE + KE
Rearranging the equation, we can solve for KE:
KE = TE - GPE
Substituting the values, we get:
KE = 11.0036 J - 3.2116 J = 7.7920 J
Therefore, the kinetic energy of the golf ball when it is 7.0 m below its highest point is 7.7920 Joules.
To find the speed at this point, we can use the formula for kinetic energy:
KE = 1/2 * m * v^2
Rearranging the formula, we can solve for v:
v = sqrt(2 * KE / m)
Substituting the values, we get:
v = sqrt(2 * 7.7920 J / 0.047 kg) = 15.83 m/s
Therefore, the speed of the golf ball when it is 7.0 m below its highest point is approximately 15.83 m/s.