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.
a) The speed of the man at the bottom of the slide is 8.45 m/s.
b) The man's kinetic energy is 545.3 J.
c) The man will travel a distance of 37.3 m before coming to rest.
(a) Well, a man sliding down a water slide sounds like a slippery situation! To find his speed at the bottom, we can use a little bit of physics. The force of gravity acting on the man can be broken down into two components: one parallel to the slide and one perpendicular to the slide. The parallel component of gravity will be opposed by the frictional force. So, the net force acting on the man will be the parallel component of gravity minus the frictional force. We can use this net force to find his acceleration (a = F_net / m). Once we have the acceleration, we can use the kinematic equation v^2 = u^2 + 2as to find the final speed (v) at the bottom, where u is the initial speed (which is 0) and s is the displacement.
So, let's do some calculations:
First, we need to find the parallel component of gravity (mg_parallel). We do this by multiplying the weight of the man (m*g) with the sine of the angle of the slide (65°).
mg_parallel = 76 kg * 9.8 m/s^2 * sin(65°)
Next, we calculate the net force:
F_net = mg_parallel - frictional force
Now, we can find the acceleration:
a = F_net / m
Finally, we can find the speed at the bottom using the kinematic equation:
v^2 = 0^2 + 2as
Keep in mind that the speed will be positive since it's moving downward.
(b) To find the kinetic energy, we can use the formula:
Kinetic energy = (1/2) * m * v^2
Just plug in the value of mass (m) and speed (v) that we found in part (a) to calculate it.
(c) To find the distance the man will travel before coming to rest, we need to know his deceleration underwater. If the frictional force remains the same, we can use the formula v^2 = u^2 - 2aD, where v is the final speed (0 m/s in this case), u is the initial speed (the speed at the bottom of the slide), a is the acceleration (opposite to the direction of motion), and D is the distance traveled. We can rearrange this equation to find D:
D = (u^2 - v^2) / (2a)
We know u (the speed at the bottom) and v (0 m/s), but we need to find the deceleration (a) first. We can do this by using the same formula as in part (a), but with the weight of the man changed in water (since underwater is a different environment).
To solve this problem, we'll use the principles of energy conservation and Newton's laws of motion. Let's break it down step by step:
Step 1: Determine the gravitational force acting on the man.
The gravitational force acting on the man can be calculated using the formula: F_gravity = m * g, where m is the mass of the man and g is the acceleration due to gravity (approximately 9.8 m/s²).
F_gravity = 76 kg * 9.8 m/s² = 745.6 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 found using the formula: F_parallel = F_gravity * sin(θ), where θ is the angle of inclination of the slide.
F_parallel = 745.6 N * sin(65°) = 650.1 N
Step 3: Calculate the net force acting on the man.
The net force is the difference between the parallel component of gravity and the frictional force: F_net = F_parallel - F_friction.
F_friction = 45.0 N
F_net = 650.1 N - 45.0 N = 605.1 N
Step 4: Calculate the acceleration of the man.
The acceleration of the man can be found using Newton's second law: F_net = m * a, where F_net is the net force and m is the mass of the man.
605.1 N = 76 kg * a
a = 7.96 m/s²
Step 5: Calculate the speed of the man at the bottom of the slide.
We can use the kinematic equation: v² = u² + 2 * a * s, where v is the final velocity, u is the initial velocity (0 m/s since he starts from rest), a is the acceleration, and s is the distance traveled (12.0 m in this case).
v² = 0² + 2 * 7.96 m/s² * 12.0 m
v² = 191.04 m²/s²
v = √191.04 m/s
v ≈ 13.84 m/s
(a) The speed of the man at the bottom of the slide is approximately 13.84 m/s.
Step 6: Calculate the kinetic energy of the man.
The kinetic energy can be determined using the equation: KE = 0.5 * m * v², where KE is the kinetic energy, m is the mass of the man, and v is the velocity.
KE = 0.5 * 76 kg * (13.84 m/s)²
KE ≈ 7675.57 J
(b) The kinetic energy of the man is approximately 7675.57 J.
Step 7: Calculate the distance traveled underwater.
To calculate the distance traveled underwater until he comes to rest, we can use the following equation: v² = u² + 2 * a * s, where v is the final velocity (0 m/s when he comes to rest), u is the initial velocity (13.84 m/s), a is the acceleration due to the frictional force, and s is the distance traveled.
0² = (13.84 m/s)² + 2 * (-45.0 N / 76 kg) * s
0 = 191.04 m²/s² - 590 N/m * s
590 N/m * s = 191.04 m²/s²
s = 191.04 m²/s² / 590 N/m
s ≈ 0.32 m
(c) The man will travel approximately 0.32 m underwater before coming to rest.
To find the answers to the given questions, we need to analyze the forces acting on the man as he slides down the water slide. Let's break down the problem step by step.
(a) What is the speed of the man at the bottom of the slide?
To determine the speed of the man at the bottom of the slide, we need to consider the work-energy principle. It states that the work done on an object is equal to the change in its kinetic energy.
The work done on the man is equal to the net work done on him.
Net work (W_net) = Work done by gravity (W_gravity) + Work done by friction (W_friction)
W_gravity = m * g * h, where m is the mass, g is the acceleration due to gravity (9.8 m/s²), and h is the height of the slide (12 m).
W_friction = F_friction * d, where F_friction is the frictional force (45.0 N) and d is the distance traveled (12 m).
W_net = W_gravity + W_friction
Now, let's calculate the net work done on the man.
W_gravity = (76 kg) * (9.8 m/s²) * (12 m) = 8928 J
W_friction = (45.0 N) * (12 m) = 540 J
W_net = 8928 J + 540 J = 9468 J
The net work done on the man is 9468 J. This is equal to the change in his kinetic energy.
ΔKE = W_net = 9468 J
The final kinetic energy (KE_final) is equal to the initial kinetic energy (KE_initial) plus the change in kinetic energy (ΔKE), and the initial kinetic energy is zero since the man starts from rest.
KE_final = KE_initial + ΔKE
KE_final = 0 + 9468 J = 9468 J
We can now calculate the speed of the man at the bottom of the slide using the formula for kinetic energy:
KE_final = (1/2) * m * v², where v is the final velocity.
9468 J = (1/2) * (76 kg) * v²
v² = (2 * 9468 J) / (76 kg)
v² = 248.8421 m²/s²
v ≈ 15.7750 m/s
Therefore, the speed of the man at the bottom of the slide is approximately 15.7750 m/s.
(b) What is his kinetic energy?
We have already calculated the kinetic energy in part (a):
KE_final = 9468 J
Therefore, his kinetic energy is 9468 J.
(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.
To find the distance the man will travel before coming to rest, we need to use the work-energy principle again.
The work done on the man is equal to the negative work done by friction underwater because it acts opposite to the direction of motion.
Net work (W_net) = Work done by friction underwater (W_friction_underwater)
W_friction_underwater = F_friction * d_underwater, where F_friction is the same as the frictional force on the slide (45.0 N) and d_underwater is the distance traveled underwater.
W_net = W_friction_underwater
W_net = -F_friction * d_underwater
We can equate this to the change in kinetic energy to find the distance traveled underwater.
W_net = ΔKE = KE_initial - KE_final
Since the initial kinetic energy is 9468 J (as found in part (b)), and the final kinetic energy is zero when he comes to rest, we have:
W_net = 9468 J - 0 = 9468 J
Therefore, we have:
9468 J = - (45.0 N) * d_underwater
Solving for d_underwater:
d_underwater = - 9468 J / (45.0 N)
d_underwater ≈ -210.4 m
The negative sign indicates that the distance underwater is in the opposite direction of the initial slide. Therefore, the man will travel approximately 210.4 meters underwater before coming to rest (assuming the frictional force remains the same).