A 76 kg man starts from rest at the top of a 12.0-m long water slide that is inclined 65° with the horizontal. In sliding down, he encountered a frictional force of 45.0 N.
(a) What is the speed of the man at the bottom of the slide?
(b) What is his kinetic energy?
(c) Suppose the man landed horizontally into the water, how far will he travel before coming to rest? Assume that the frictional force is the same underwater.
Put your solutions
(a) The speed of the man at the bottom of the slide is 8.45 m/s.
(b) His kinetic energy is 545.3 J.
(c) The man will travel a distance of 37.3 m before coming to rest.
Oh boy, we're sliding into some physics here! But fear not, I, Clown Bot, will try to make it more fun. Let's get cracking!
(a) To find the speed of the man at the bottom of the slide, we need to use some energy conservation. When he starts from rest, all his potential energy turns into kinetic energy at the bottom. We can calculate his potential energy using the formula: PE = mgh, where m = 76 kg, g = 9.8 m/s², and h = 12.0 m. So his potential energy is PE = (76 kg)(9.8 m/s²)(12.0 m). Now, assuming there are no energy losses due to friction, we can equate his potential energy to his kinetic energy at the bottom: PE = KE. Therefore, we can find the speed (v) using the formula: KE = 0.5mv². You can rearrange that equation to solve for v. Get ready for some numbers!
(b) Okie-dokie, now let's find his kinetic energy. We can use the formula I just mentioned: KE = 0.5mv². Plug in the mass (m) of 76 kg and the velocity (v) you found in part (a) – and voila! Kinetic energy in the bag!
(c) Now, let's dive into the distance the man will travel before coming to rest in the water. Assuming the frictional force is the same underwater, we need to find the work done by friction to bring the man to rest. The formula for work (W) is W = Fd cosθ. In this case, the frictional force (F) is given as 45.0 N, and the angle (θ) would be 180 degrees since he lands horizontally. The work done should be equal to the change in kinetic energy (KE) from part (b), which is zero (since he comes to rest). Use the work formula to solve for the distance (d) – and there you have it!
Don't forget to plug in the values and do the math, my friend. You got this!
Given information:
Mass of the man (m) = 76 kg
Length of the water slide (d) = 12.0 m
Angle of the slide (θ) = 65°
Frictional force (f) = 45.0 N
(a) To find the speed of the man at the bottom of the slide, we can start by using the conservation of energy. The initial potential energy at the top of the slide will be converted into kinetic energy at the bottom of the slide, neglecting any losses due to friction.
The potential energy at the top of the slide can be calculated using the following formula:
Potential energy (PE) = m * g * h
Where g is the acceleration due to gravity (approximately 9.8 m/s^2) and h is the vertical height.
Since the slide is inclined, we need to find the vertical height using trigonometry:
h = d * sin(θ)
Substituting the values:
h = 12.0 m * sin(65°) ≈ 10.325 m
Now we can calculate the potential energy:
PE = 76 kg * 9.8 m/s^2 * 10.325 m ≈ 7,571.032 J
The potential energy is converted to kinetic energy at the bottom of the slide, so
KE = PE
Kinetic energy (KE) = 7,571.032 J
The kinetic energy can also be calculated using the formula:
Kinetic energy (KE) = 0.5 * m * v^2
Where v is the speed at the bottom of the slide that we need to find.
Setting up the equation:
0.5 * 76 kg * v^2 = 7,571.032 J
Simplifying the equation:
38 kg * v^2 = 7,571.032 J
Dividing both sides by 38 kg:
v^2 = 199.237 J/kg
Taking the square root of both sides:
v ≈ √(199.237 J/kg) ≈ 14.11 m/s
Therefore, the speed of the man at the bottom of the slide is approximately 14.11 m/s.
(b) The kinetic energy of the man can be calculated using the formula:
Kinetic energy (KE) = 0.5 * m * v^2
Substituting the values:
KE = 0.5 * 76 kg * (14.11 m/s)^2 ≈ 7,571.032 J
Therefore, the kinetic energy of the man is approximately 7,571.032 J.
(c) To find the distance the man will travel before coming to rest in water, we can use the work-energy principle. The work done by the frictional force will be equal to the initial kinetic energy of the man, and it can be calculated as:
Work done by friction (W) = f * d
Substituting the values:
W = 45.0 N * 12.0 m ≈ 540 J
Since the work done is equal to the initial kinetic energy, we can set up the equation:
540 J = 0.5 * m * v^2
Rearranging the equation:
v^2 = (2 * 540 J) / 76 kg
v^2 ≈ 14.21 m^2/s^2
Taking the square root of both sides:
v ≈ √(14.21 m^2/s^2) ≈ 3.77 m/s
Now we can use the equation of motion to find the distance traveled before coming to rest:
v^2 = u^2 + 2 * a * s
Where u is the initial speed, a is the acceleration, and s is the distance.
Since the man comes to rest, the final speed (v) is 0 m/s. The initial speed (u) is 3.77 m/s, and the acceleration (a) can be calculated using the frictional force and mass:
a = f / m
a = 45.0 N / 76 kg ≈ 0.592 m/s^2
Substituting the values into the equation of motion:
0^2 = (3.77 m/s)^2 + 2 * 0.592 m/s^2 * s
0 = 14.1929 m^2/s^2 + 1.184 m/s^2 * s
Rearranging the equation:
1.184 m/s^2 * s = -14.1929 m^2/s^2
Simplifying:
s = -14.1929 m^2/s^2 / 1.184 m/s^2
s ≈ 12.0 m
The negative sign indicates that the direction of displacement is opposite to the initial motion. Therefore, the man will travel approximately 12.0 m underwater before coming to rest.
In summary:
(a) The speed of the man at the bottom of the slide is approximately 14.11 m/s.
(b) The kinetic energy of the man is approximately 7,571.032 J.
(c) The man will travel approximately 12.0 m underwater before coming to rest.
To solve this problem, we can break it down into several steps:
Step 1: Calculate the gravitational force acting on the man.
The gravitational force can be calculated using the formula:
Gravitational force = mass x acceleration due to gravity
Gravitational force = 76 kg x 9.8 m/s² = 744.8 N
Step 2: Calculate the component of the gravitational force parallel to the slide.
The component of the gravitational force parallel to the slide can be calculated using the formula:
Force parallel = Gravitational force x sin(angle)
Force parallel = 744.8 N x sin(65°) = 657.20 N
Step 3: Calculate the net force acting on the man.
Since the man encounters a frictional force, the net force can be calculated using the formula:
Net force = Force parallel - Frictional force
Net force = 657.20 N - 45.0 N = 612.20 N
Step 4: Calculate the acceleration of the man.
The acceleration of the man can be calculated using Newton's second law of motion:
Net force = mass x acceleration
Acceleration = Net force / mass
Acceleration = 612.20 N / 76 kg = 8.06 m/s²
Step 5: Calculate the speed of the man at the bottom of the slide.
The final speed of the man can be calculated using the equation of motion:
Final speed² = Initial speed² + 2 x acceleration x distance
Since the man starts from rest (initial speed = 0 m/s), the equation simplifies to:
Final speed = sqrt(2 x acceleration x distance)
Final speed = sqrt(2 x 8.06 m/s² x 12.0 m) = 7.28 m/s
(a) The speed of the man at the bottom of the slide is 7.28 m/s.
Step 6: Calculate the kinetic energy of the man.
The kinetic energy can be calculated using the formula:
Kinetic energy = 0.5 x mass x (velocity)²
Kinetic energy = 0.5 x 76 kg x (7.28 m/s)² = 2032.65 J
(b) The kinetic energy of the man is 2032.65 J.
Step 7: Calculate the distance the man will travel before coming to rest underwater.
Since the frictional force is the same underwater, we can use the same net force from Step 3 to calculate the distance.
The distance can be calculated using the equation of motion:
Final speed² = Initial speed² + 2 x acceleration x distance
Since the final speed is 0 m/s (coming to rest), the equation simplifies to:
distance = (Initial speed²) / (2 x acceleration)
distance = (7.28 m/s)² / (2 x 8.06 m/s²) = 2.07 m
(c) The man will travel approximately 2.07 meters before coming to rest underwater.