You are pulling your sister on a sled to the top of a 20.0 m high, frictionless hill with a 10.0° incline. Your sister and the sled have a total mass of 50.0 kg. You pull the sled, starting from rest, with a constant force of 127 N at an angle of 45.0° to the hill. If you pull from the bottom to the top, what will the speed of the sled be when you reach the top? answer in m/s.
well, the pulling force in the direction up the hill is 127N*cos45.
KE at top=127Cos45*20/sin10 -50*g*20
solve for veloicyt from KE.
To find the speed of the sled when you reach the top of the hill, we can use the principle of conservation of energy.
Step 1: Calculate the gravitational potential energy at the top of the hill.
The gravitational potential energy (PE) is given by the equation:
PE = mgh
where m is the mass, g is the acceleration due to gravity, and h is the height.
The mass of your sister and the sled combined is 50.0 kg.
The height of the hill is 20.0 m.
The acceleration due to gravity (g) is approximately 9.8 m/s^2.
PE = 50.0 kg × 9.8 m/s^2 × 20.0 m
PE = 9800 J
Step 2: Calculate the work done against gravity.
The work done against gravity (W) is given by the equation:
W = Fd
where F is the force applied and d is the distance over which the force is applied.
The force applied is 127 N and the distance is the length of the hill along the incline. To find the distance, we can use the trigonometric relationship between the angle of inclination and the height of the hill along the incline.
sin(angle) = opposite/hypotenuse
sin(10.0°) = height_along_incline/20.0 m
Solving for the height_along_incline:
height_along_incline = sin(10.0°) × 20.0 m
height_along_incline ≈ 3.46 m
Therefore, the distance over which the force is applied is 3.46 m.
W = 127 N × 3.46 m
W ≈ 439 J
Step 3: Calculate the kinetic energy at the top of the hill.
Using the principle of conservation of energy, the total energy (KE + PE) at the top must be equal to the total energy (KE + PE) at the bottom, assuming no frictional losses.
At the bottom of the hill, the sled starts from rest, so the initial kinetic energy (KE) is zero.
KE_bottom + PE_bottom = KE_top + PE_top
0 + 0 = KE_top + 9800 J
Simplifying the equation:
KE_top = -9800 J
Step 4: Calculate the final velocity.
The final velocity (v) can be found using the formula:
KE = (1/2)mv^2
Substituting the known values:
-9800 J = (1/2) × 50.0 kg × v^2
Simplifying the equation:
v^2 = -9800 J × 2 / 50.0 kg
v^2 ≈ -392 m^2/s^2
Since velocity cannot be negative, the final velocity (v) is zero m/s.
Therefore, the speed of the sled when you reach the top of the hill is zero m/s.
To find the speed of the sled when you reach the top of the hill, we need to consider the work done against gravity and the work done by the applied force.
1. First, let's find the work done against gravity.
The gravitational potential energy (PE) at the starting point and the end point of the sled can be calculated using the formula:
PE = m * g * h
where m is the total mass, g is the acceleration due to gravity, and h is the vertical height.
PE_start = m * g * h_start = 50.0 kg * 9.8 m/s² * 0 m = 0 J (since the sled starts from rest at the bottom).
PE_end = m * g * h_end = 50.0 kg * 9.8 m/s² * 20.0 m = 9800 J
The difference in potential energy is the work done against gravity:
W_gravity = PE_end - PE_start = 9800 J - 0 J = 9800 J
2. Next, let's find the work done by the applied force.
The work done by a force can be calculated using the equation:
W_applied = F * d * cos(θ)
where F is the magnitude of the applied force, d is the displacement, and θ is the angle between the force and the displacement.
In this case, the angle θ is 45.0°, the magnitude of the applied force F is 127 N, and the displacement d can be calculated as follows:
d = h / sin(α)
where α is the angle of the incline. In this case, α is 10.0°.
d = 20.0 m / sin(10.0°) = 116.21 m
Plugging in the values:
W_applied = 127 N * 116.21 m * cos(45.0°) = 8084.17 J
3. Finally, let's find the work-energy theorem to find the final kinetic energy and then use it to calculate the speed of the sled when you reach the top.
According to the work-energy theorem, the work done by all the forces acting on an object is equal to the change in its kinetic energy (KE).
W_net = ΔKE
Since there are no other forces acting, the net work done is equal to the sum of the work done against gravity and the work done by the applied force:
W_net = W_gravity + W_applied = 9800 J + 8084.17 J = 17884.17 J
ΔKE = 17884.17 J
The final kinetic energy can be calculated using the formula:
KE = 0.5 * m * v²
where m is the total mass and v is the final velocity.
0.5 * 50.0 kg * v² = 17884.17 J
Solving for v:
v² = (17884.17 J) / (0.5 * 50.0 kg) = 717.68 m²/s²
v = √(717.68 m²/s²) ≈ 26.79 m/s
Therefore, the speed of the sled when you reach the top of the hill is approximately 26.79 m/s.