sorry here is actual question:
A 285-kg stunt boat is driven on the surface of a lake at a constant speed of 13.5 m/s toward a ramp, which is angled at 28.5° above the horizontal. The coefficient of friction between the boat bottom and the ramp's surface is 0.150, and the raised end of the ramp is 1.70 m above the water surface.
a. speed of boat when leaves ramp (engine off once hits ramp)
b. speed of boat when hits water again (no air resistance)
The work-energy theorem
KE2 = KE1 –W(fr) –PE,
KE2 = m•v1²/2 - k•m•g•cosα•s - m•g•h =
= m•v²/2 - m•g•h{(k/tanα) +1} = 2•10^4 J.
KE2 = m•v2²/2,
v2 = sqrt(2•KE2/m) = 11.8 m/s,
The work-kinetic energy theorem for the
boat while it is airborne:
KE3 = ΔPE+KE2 = m•g•h+ m•v2²/2 = 2.46•10^4 J.
KE3 = m•v3²/2,
v3 = sqrt(2•KE3/m) = 13.17 m/s.
To solve this problem, we will use the principle of conservation of mechanical energy. We can break down the problem into two parts: before the boat leaves the ramp and after the boat hits the water again. Let's solve it step by step.
Step 1: Calculate the potential energy of the boat at the raised end of the ramp.
The potential energy (PE) of the boat can be calculated using the formula:
PE = m * g * h
Where:
m = mass of the boat = 285 kg
g = acceleration due to gravity = 9.8 m/s²
h = height of the ramp = 1.70 m
Substituting the values:
PE = 285 kg * 9.8 m/s² * 1.70 m
PE ≈ 4785.3 J
Step 2: Calculate the initial kinetic energy of the boat before it leaves the ramp.
The initial kinetic energy (KE) of the boat can be calculated using the formula:
KE = (1/2) * m * v²
Where:
m = mass of the boat = 285 kg
v = velocity of the boat = 13.5 m/s
Substituting the values:
KE = (1/2) * 285 kg * (13.5 m/s)²
KE ≈ 12304.5 J
Step 3: Calculate the work done by friction on the boat as it moves up the ramp.
The work done by friction (W) can be calculated using the formula:
W = friction force * distance
The friction force can be determined using the equation:
friction force = coefficient of friction * normal force
The normal force can be calculated using the formula:
normal force = m * g * cos(θ)
Where:
m = mass of the boat = 285 kg
g = acceleration due to gravity = 9.8 m/s²
θ = angle of the ramp = 28.5°
First, calculate the normal force:
normal force = 285 kg * 9.8 m/s² * cos(28.5°)
normal force ≈ 2452.9 N
Now, calculate the friction force:
friction force = 0.150 * 2452.9 N
friction force ≈ 367.9 N
Since the boat is moving up the ramp, the work done by friction is negative:
W = - friction force * distance
However, the distance over which the force is applied is not given. Therefore, we cannot calculate the exact work done by friction in this case.
Step 4: Calculate the final kinetic energy of the boat after it leaves the ramp.
Since there is no work done by friction once the boat leaves the ramp, the mechanical energy is conserved. Therefore, the final kinetic energy (KE') is equal to the initial kinetic energy (KE) plus the potential energy (PE):
KE' = KE + PE
Substituting the values:
KE' ≈ 12304.5 J + 4785.3 J
KE' ≈ 17089.8 J
Step 5: Calculate the final velocity of the boat after it leaves the ramp.
The final velocity (v') can be calculated using the formula:
KE' = (1/2) * m * v'²
Rearranging the formula:
v'² = (2 * KE') / m
v' = √((2 * KE') / m)
Substituting the values:
v' ≈ √((2 * 17089.8 J) / 285 kg)
v' ≈ √119.65 m²/s²
v' ≈ 10.95 m/s
Therefore, the speed of the boat when it leaves the ramp (with the engine off) is approximately 10.95 m/s.
To calculate the speed of the boat when it hits the water again (with no air resistance), we need to use the principle of conservation of mechanical energy again.
Step 6: Calculate the final potential energy of the boat when it hits the water.
The final potential energy (PE") of the boat can be calculated using the formula:
PE" = m * g * h
Where:
m = mass of the boat = 285 kg
g = acceleration due to gravity = 9.8 m/s²
h = height of the ramp = 1.70 m
Substituting the values:
PE" = 285 kg * 9.8 m/s² * 1.70 m
PE" ≈ 4785.3 J
Step 7: Calculate the final kinetic energy of the boat when it hits the water.
Since there is no work done by friction or any other force, the final kinetic energy (KE'') is equal to the initial kinetic energy (KE') plus the final potential energy (PE"):
KE'' = KE' + PE"
Substituting the values:
KE'' ≈ 17089.8 J + 4785.3 J
KE'' ≈ 21875.1 J
Step 8: Calculate the final velocity of the boat when it hits the water.
The final velocity (v'') can be calculated using the formula:
KE'' = (1/2) * m * v''²
Rearranging the formula:
v''² = (2 * KE'') / m
v'' = √((2 * KE'') / m)
Substituting the values:
v'' ≈ √((2 * 21875.1 J) / 285 kg)
v'' ≈ √153.5 m²/s²
v'' ≈ 12.4 m/s
Therefore, the speed of the boat when it hits the water again (with no air resistance) is approximately 12.4 m/s.
To solve this problem, we need to analyze the forces acting on the boat, apply Newton's laws of motion, and use some basic trigonometry. Here's how you can find the answers:
a. To determine the speed of the boat when it leaves the ramp, we first need to calculate the normal force and the frictional force acting on the boat.
1. Calculate the normal force: The normal force is the force exerted by the ramp perpendicular to its surface. In this case, it counterbalances the downward force of gravity acting on the boat.
Normal force = mass × acceleration due to gravity × cos(angle of the ramp)
N = 285 kg × 9.8 m/s^2 × cos(28.5°)
2. Calculate the frictional force: The frictional force acts parallel to the ramp's surface and opposes the boat's motion.
Frictional force = coefficient of friction × normal force
F_friction = 0.150 × N
3. Calculate the net force on the boat: Since it moves at a constant speed, the net force on the boat must be zero.
Net force = 0
Net force = Thrust force − Frictional force − Component of gravity parallel to the ramp
For the component of gravity parallel to the ramp:
Component of gravity parallel to the ramp (F_gravity_parallel) = mass × acceleration due to gravity × sin(angle of the ramp)
F_gravity_parallel = 285 kg × 9.8 m/s^2 × sin(28.5°)
Now that we know the frictional force, we can find the thrust force:
Thrust force = Frictional force + Component of gravity parallel to the ramp
4. Finally, using the net force, we can calculate the boat's acceleration:
Net force = mass × acceleration
acceleration = net force / mass
acceleration = 0 / 285 kg
Since the net force is zero, the boat experiences no acceleration, meaning its speed remains constant. Therefore, the speed of the boat when it leaves the ramp is 13.5 m/s.
b. To find the speed of the boat when it hits the water again, we need to calculate the vertical distance the boat travels while in the air.
1. Calculate the time the boat is in the air: The boat will be in the air while it travels the vertical distance between the ramp and the water's surface. We can use the equation of motion in vertical direction:
Vertical distance = Initial vertical velocity × time + (0.5 × acceleration due to gravity × time^2)
The initial vertical velocity is zero as the boat leaves the ramp. Rearrange the equation to solve for time:
t = sqrt((2 × vertical distance) / acceleration due to gravity)
t = sqrt((2 × 1.70 m) / 9.8 m/s^2)
2. Calculate the horizontal distance traveled by the boat in the air: This can be found using the horizontal component of the boat's velocity:
horizontal distance = (horizontal velocity) × time
The horizontal velocity remains constant at 13.5 m/s.
horizontal distance = 13.5 m/s × t
3. Finally, calculate the boat's speed when it hits the water:
The total distance traveled by the boat is the sum of the horizontal distance and the vertical distance.
total distance = horizontal distance + vertical distance
average speed = total distance / time
speed = average speed
By substituting the values, you can calculate the speed of the boat when it hits the water again.
Remember to double-check your calculations and unit conversions.