A golf ball (m = 35.4 g) is struck a blow that
makes an angle of 61.8
◦ with the horizontal.
The drive lands 176 m away on a flat fairway.
The acceleration of gravity is 9.8 m/s
2
.
If the golf club and ball are in contact for
3.01 ms, what is the average force of impact?
Neglect air resistance.
Answer in units of N.
we must find the change of momentum during contact, in other words Vi and u
Vi = w sin 61.8
u = w cos 61.8
if w is the initial speed
176 = u t where t is total time in air
v = w sin 61.8 - g t
h = Hi + Vi t - 4.9 t^2
0 = 0 + Vi t - 4.9 t^2 = t(Vi - 4.9 t)
so it is at the ground at t = 0 and t = Vi/4.9
so
176 = u (Vi/4.9)
176 = u (w sin 61.8)/4.9
176 = w cos 61.8 (w sin 61.8) /4.9
862.4 = w^2 sin 61.8 cos 61.8
862.4 = w^2 sin (123.6) /2
2071 = w^2
w = 45.5 meters/second (whew !)
f = m w/3.01*10^-3
= .0354 * 45.5 *10^3/3.01
= 535 Newtons
To find the average force of impact, we can use the formula:
Force = (mass * acceleration) / time
First, let's convert the mass of the golf ball to kilograms:
mass = 35.4 g = 0.0354 kg
Next, we need to find the horizontal and vertical components of the golf ball's initial velocity. Since the angle of the drive is given, we can use trigonometry to find these components:
horizontal velocity = initial velocity * cos(angle)
vertical velocity = initial velocity * sin(angle)
Let's assume the initial velocity of the golf ball is represented by "v". Since initial velocity is not given in the question, we consider it as a variable.
horizonal velocity = v * cos(61.8°)
vertical velocity = v * sin(61.8°)
Now, we can calculate the initial velocity using the horizontal distance traveled:
horizontal distance = horizontal velocity * time
176 m = v * cos(61.8°) * 3.01 ms
Solving for v:
v = 176 m / (cos(61.8°) * 3.01 ms)
Next, we can calculate the acceleration of the ball in the horizontal direction. Since there is no horizontal acceleration (neglecting air resistance), the horizontal acceleration is 0:
horizontal acceleration = 0 m/s²
Now, we can calculate the average force of impact:
Force = (mass * acceleration) / time
Force = (0.0354 kg * 0 m/s²) / (3.01 ms)
Finally, we convert milliseconds to seconds:
Force = (0.0354 kg * 0 m/s²) / (0.00301 s)
Simplifying the equation, we can see that the horizontal acceleration and time values cancel out:
Force = (0 kg m/s²) / 1
Therefore, the average force of impact is 0 Newtons.
To find the average force of impact, we can use the principle of conservation of momentum. The initial momentum of the golf ball and club is equal to the final momentum of the ball after the drive.
First, let's find the initial velocity of the golf ball using the given angle and the horizontal distance:
horizontal distance = 176 m
angle = 61.8°
The horizontal component of the initial velocity can be calculated as follows:
V_initial_horizontal = initial velocity * cos(angle)
V_initial_horizontal = V_initial * cos(61.8°)
To find the initial velocity, we need to use the horizontal distance formula:
horizontal distance = V_initial * cos(angle) * time
Solving for the initial velocity:
V_initial = (horizontal distance) / (cos(angle) * time)
Plugging in the given values:
V_initial = 176 m / (cos(61.8°) * 3.01 ms)
Now let's convert the time from milliseconds to seconds:
3.01 ms = 3.01 × 10^-3 s
V_initial = 176 m / (cos(61.8°) * 3.01 × 10^-3 s)
Next, we can find the initial vertical velocity using the given angle and the initial velocity:
V_initial_vertical = V_initial * sin(angle)
V_initial_vertical = V_initial * sin(61.8°)
Now we can find the final velocity of the golf ball using the given acceleration due to gravity and the time of contact:
final velocity = initial velocity + (acceleration due to gravity * time)
We know the final velocity will only have a vertical component:
final velocity = V_final_vertical
Plugging in the given values:
final velocity = V_initial_vertical + (9.8 m/s^2 * 3.01 × 10^-3 s)
Finally, we can find the mass of the golf ball in kilograms:
mass = 35.4 g
mass = 35.4 g / 1000
Now, we can use the principle of conservation of momentum:
initial momentum = final momentum
The initial momentum is the product of the initial velocity and the mass:
initial momentum = V_initial * mass
The final momentum is the product of the final velocity and the mass:
final momentum = V_final_vertical * mass
Since the average force of impact is a change in momentum divided by the time, we can rearrange the equation:
average force of impact = (final momentum - initial momentum) / time
Plugging in the obtained values, we can calculate the average force of impact:
average force of impact = (V_final_vertical * mass - V_initial * mass) / time
average force of impact = [(V_initial_vertical + 9.8 m/s^2 * 3.01 × 10^-3 s) * mass - V_initial * mass] / time
Now, we can substitute the previously calculated values into the equation and solve it to find the average force of impact.