An 80 kg stuntman jumps from the top of a building 23 m above a catching net. Assuming that air resistance exerts a 100 N force on the stuntman as he falls, determine his velocity just before he hits the net.
(m)(g)(h)-(F)(d)=(1/2)(m)(v2^2)
mgh-(F)(d)=(1/2)(m)(v^2)
(80)(9.8)(23)-(100)(23)=(1/2)(80)(v^2)
18032-2300=(40)(v^2)
15732=(40)(v^2)
393.3=(v^2)
v=19.832 m/s
Well, let's do some calculations and hope no stuntmen get hurt in the process! So, we'll start with the good ol' Newton's second law of motion, which states that the force applied to an object is equal to its mass multiplied by its acceleration.
The force on the stuntman is given as the force due to gravity minus the force of air resistance. The force due to gravity can be calculated by multiplying the mass of the stuntman by the acceleration due to gravity (9.8 m/s^2). So, that comes out to be 80 kg × 9.8 m/s^2 = 784 N.
Now, let's calculate the acceleration acting on the stuntman. Since force and acceleration are directly proportional, we divide the net force by the mass of the stuntman. In this case, the net force is 784 N - 100 N = 684 N. So, acceleration (a) = net force (F) / mass (m) = 684 N / 80 kg = 8.55 m/s^2.
Using the kinematic equation Vf^2 = Vi^2 + 2ad, where Vf is the final velocity, Vi is the initial velocity (which is 0 in this case), a is acceleration, and d is the displacement, we can solve for Vf.
Plugging in the values, Vf^2 = 0 + 2 × 8.55 m/s^2 × 23 m = 393.9. Taking the square root, Vf = √393.9 = 19.8 m/s.
So, at the moment our intrepid stuntman hits the net, his velocity is approximately 19.8 m/s. Just remember, he's a professional, so don't try this at home!
To determine the velocity of the stuntman just before he hits the net, we will use the laws of motion.
Step 1: Calculate the net force acting on the stuntman.
The net force is the sum of the gravitational force and the force due to air resistance.
Gravitational force (weight) = mass x acceleration due to gravity
Weight = 80 kg x 9.8 m/s^2 (acceleration due to gravity)
Weight = 784 N
Net force = weight - force due to air resistance
Net force = 784 N - 100 N
Net force = 684 N
Step 2: Use Newton's second law of motion to calculate acceleration.
Newton's second law states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
Net force = mass x acceleration
684 N = 80 kg x acceleration
acceleration = 684 N / 80 kg
acceleration = 8.55 m/s^2
Step 3: Use kinematic equation to calculate velocity.
The kinematic equation relating velocity, initial velocity, acceleration, and displacement is:
v^2 = u^2 + 2as
Where:
v = final velocity (which we will determine)
u = initial velocity (0 m/s since the stuntman jumps from rest)
a = acceleration (which we calculated as 8.55 m/s^2)
s = displacement (23 m, as the stuntman falls from the top of the building to the net)
Plugging in the values:
v^2 = 0^2 + 2(8.55 m/s^2) (23 m)
v^2 = 0 + 394.65 m^2/s^2
v = √394.65 m^2/s^2
v ≈ 19.87 m/s
Therefore, the velocity of the stuntman just before he hits the net is approximately 19.87 m/s.
To determine the velocity of the stuntman just before he hits the net, we can use the concept of conservation of energy.
Let's break down the problem into several steps:
Step 1: Calculate the potential energy at the top of the building.
Potential energy (PE) is given by the equation PE = mass × gravity × height.
In this case, the mass of the stuntman is 80 kg, the acceleration due to gravity is approximately 9.8 m/s², and the height of the building is 23 m.
Therefore, the potential energy at the top of the building is:
PE = 80 kg × 9.8 m/s² × 23 m.
Step 2: Account for the work done by air resistance.
Since air resistance exerts a force on the stuntman, work is done against this force. Work (W) is given by the equation W = force × distance.
In this case, the force of air resistance is 100 N and the distance is 23 m.
Therefore, the work done by air resistance is:
W = 100 N × 23 m.
Step 3: Calculate the kinetic energy just before hitting the net.
The net is at a height of 0 m, so all of the potential energy is converted into kinetic energy. The total mechanical energy (ME) remains constant, given by the equation ME = PE + KE, where KE represents kinetic energy.
Therefore, the kinetic energy just before hitting the net is equal to the total mechanical energy:
KE = (PE + W).
Step 4: Determine the velocity.
The formula to calculate kinetic energy is KE = (1/2) × mass × velocity².
Setting the equation from Step 3 equal to the equation for kinetic energy:
(1/2) × mass × velocity² = (PE + W).
Rearranging the equation and solving for velocity:
velocity = sqrt((2 × (PE + W)) / mass).
Now we can plug in the given values and calculate the velocity: