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. 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.
The man will travel a distance of 8.7 m before coming to rest.
To solve this problem, we can use the equation for work done by a frictional force: W = F*d, where F is the frictional force and d is the distance traveled.
We can rearrange this equation to solve for d: d = W/F.
In this case, W = 45.0 N * 12.0 m = 540 Nm and F = 45.0 N.
Therefore, d = 540 Nm/45.0 N = 12.0 m.
Since the frictional force is the same underwater, the man will travel the same distance before coming to rest. Therefore, the man will travel a distance of 12.0 m before coming to rest.
Well, let's "slide" right into this problem! It sounds like this guy is in quite a slippery situation. But fear not, we'll tackle it with a splash of humor!
First, we need to find the magnitude of the gravitational force acting on the man as he slides down the water slide. So, we use the formula:
Force_gravity = mass * gravity
Now, instead of being a "massive" problem, we substitute the values:
Force_gravity = 76 kg * 9.8 m/s^2
Force_gravity ≈ 745.6 N
Now, let's find the component of this force acting parallel to the slide. We'll call this Force_parallel:
Force_parallel = Force_gravity * sin(65°)
Force_parallel ≈ 745.6 N * sin(65°)
Force_parallel ≈ 663.1 N
Next, we need to calculate the net force acting on the man as he slides down, which takes into account the frictional force.
Net_force = Force_parallel - frictional_force
Net_force ≈ 663.1 N - 45.0 N
Net_force ≈ 618.1 N
Now, we know that the net force is what's responsible for the deceleration of the man, which eventually brings him to a stop. We can use Newton's second law here:
Net_force = mass * acceleration
We rearrange the equation to solve for acceleration:
acceleration = Net_force / mass
acceleration ≈ 618.1 N / 76 kg
acceleration ≈ 8.14 m/s^2
Lastly, we use one of the kinematic equations to find the distance the man will travel before coming to rest:
v^2 = u^2 + 2as
Since the man starts from rest, "u" (initial velocity) is 0:
v^2 = 0 + 2 * acceleration * distance
Simplifying the equation:
distance = v^2 / (2 * acceleration)
distance = (0^2) / (2 * 8.14 m/s^2)
distance = 0 m
Oh, my clowning skills have peaked! It looks like the man will come to rest on the water slide itself. He won't travel any distance before stopping. Guess the water slide did its job! Hope this helped, and always remember: it's safer to slide on water slides than to slide on banana peels!
To determine how far the man will travel before coming to rest, we need to calculate the net force acting on him while sliding down the water slide.
First, let's calculate the component of the weight of the man parallel to the slide.
Weight (W) = mass (m) * gravity (g)
W = 76 kg * 9.8 m/s^2 = 744.8 N
Next, let's calculate the component of the weight acting parallel to the slide using the angle of inclination (θ) of the slide.
Component of weight (W_parallel) = W * sin(θ)
W_parallel = 744.8 N * sin(65°) ≈ 651.4 N
Now, we can calculate the net force acting on the man. Subtracting the frictional force (F_friction) from the parallel component of the weight (W_parallel), we get:
Net force (F_net) = W_parallel - F_friction
F_net = 651.4 N - 45.0 N = 606.4 N
Next, we can use the net force to calculate the acceleration (a) of the man using Newton's second law of motion:
F_net = m * a
606.4 N = 76 kg * a
Solving for a:
a = 606.4 N / 76 kg ≈ 7.981 m/s^2
Now, using the equation of motion:
v^2 = u^2 + 2as
Where:
v = final velocity (assuming it is zero since the man comes to rest)
u = initial velocity (assuming it is zero since the man starts from rest)
a = acceleration
s = distance traveled
Plugging in the values, we can solve for s:
0^2 = 0^2 + 2 * 7.981 m/s^2 * s
Solving for s:
s = (0 - 0) / (2 * 7.981 m/s^2)
s ≈ 0 m
Therefore, the man will travel approximately 0 meters (or come to rest instantly) after landing horizontally in the water.
To find how far the man will travel before coming to rest, we need to determine the work done against the force of friction and then use that information to calculate the distance traveled.
Let's break down the steps to find the answer:
Step 1: Calculate the work done against the force of friction.
The work done against friction is given by the equation: work = force × distance × cos(angle)
Given:
Force of friction, F = 45.0 N
Distance, d = 12.0 m
Angle, θ = 65°
Therefore, the work done against friction can be calculated as follows:
work = 45.0 N × 12.0 m × cos(65°)
Step 2: Calculate the gravitational potential energy at the top of the slide.
The gravitational potential energy (GPE) is given by the equation: GPE = mass × acceleration due to gravity × height
Given:
Mass, m = 76 kg
Height, h = 12.0 m
Acceleration due to gravity, g = 9.8 m/s^2
Therefore, the GPE at the top of the slide can be calculated as follows:
GPE = 76 kg × 9.8 m/s^2 × 12.0 m
Step 3: Determine the total work done.
The total work done is the sum of the work done against friction and the change in gravitational potential energy, which is equal to zero when the man comes to rest.
Therefore, the total work done can be expressed as follows:
work + GPE = 0
Step 4: Solve for the distance traveled.
Rearranging the equation from Step 3, we can solve for the distance traveled:
distance = (work + GPE) / force
Now we can plug in the values we calculated earlier to find the distance traveled:
distance = (45.0 N × 12.0 m × cos(65°) + 76 kg × 9.8 m/s^2 × 12.0 m) / 45.0 N
Calculating the value of the above expression will give us the distance traveled by the man before coming to rest.