A 45 kg child sits on a 5 kg sled and slides down a 117 meter, 31 degree slope, to the nearest m/s what is his or her speed at the bottom?
h = 117*sin 31 = 60.26 m.
V^2 = Vo^2 + 2g*h
V^2 = 0 + 19.6*60.26 = 1181.09
V = 34.37 m/s
To find the speed of the child at the bottom of the slope, we can use the principles of physics, specifically considering the forces acting on the child-sled system.
Step 1: Calculate the gravitational force acting on the child-sled system.
The gravitational force can be calculated using the formula:
F_gravity = mass × gravity
where mass is the combined mass of the child and the sled, and gravity is the acceleration due to gravity (approximately 9.8 m/s²).
Child's mass: 45 kg
Sled's mass: 5 kg
Combined mass: 45 kg + 5 kg = 50 kg
F_gravity = 50 kg × 9.8 m/s²
F_gravity = 490 N
Step 2: Decompose the gravitational force vector into parallel and perpendicular components.
Since the slope is inclined at an angle of 31 degrees, we need to decompose the force into its parallel and perpendicular components. The perpendicular component will not affect the motion, so we only need the parallel component.
Parallel component of F_gravity = F_gravity × sin θ
Parallel component of F_gravity = 490 N × sin(31°)
Parallel component of F_gravity = 490 N × 0.515
Parallel component of F_gravity ≈ 252.85 N (rounded to two decimal places)
Step 3: Calculate the net force acting on the child-sled system.
The net force is the force that causes the child-sled system to accelerate down the slope. It can be calculated using the formula:
Net force = m × a
where m is the mass of the child-sled system and a is the acceleration.
Since the system is accelerating down the slope, the net force is equal to the parallel component of the gravitational force.
Net force = 252.85 N
Step 4: Calculate the acceleration of the child-sled system.
The acceleration can be calculated using Newton's second law of motion:
Net force = m × a
Rearranging the formula, we get:
a = Net force / m
a = 252.85 N / 50 kg
a ≈ 5.06 m/s² (rounded to two decimal places)
Step 5: Calculate the speed of the child at the bottom of the slope.
To calculate the speed, we can use the kinematic equation:
v² = u² + 2as
where v is the final velocity (speed), u is the initial velocity (which we can assume is 0 since the child starts from rest), a is the acceleration, and s is the distance traveled down the slope.
Initial velocity (u) = 0 m/s
Distance traveled down the slope (s) = 117 m
Acceleration (a) = 5.06 m/s²
Plugging in the values:
v² = 0² + 2 × 5.06 m/s² × 117 m
v² = 0 + 2 × 5.06 m/s² × 117 m
v² = 0 + 2 × 591.42 m²/s²
v² = 1182.84 m²/s²
Taking the square root of both sides:
v ≈ √(1182.84 m²/s²)
v ≈ √1182.84 m/s
v ≈ 34.37 m/s (rounded to two decimal places)
Therefore, the child's speed at the bottom of the slope is approximately 34.37 m/s.
To find the speed of the child at the bottom of the slope, we can use the principle of conservation of energy. The energy at the top of the slope can be converted to the energy at the bottom of the slope.
We can consider the initial energy as potential energy. The potential energy is given by the formula P.E. = mgh, where m is the mass (in kg), g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height of the slope (in meters). In this case, the height of the slope is the vertical distance traveled, which can be calculated using trigonometry.
The formula for the vertical distance is given by h = d * sin(theta), where d is the length of the slope (in this case, 117 meters) and theta is the angle of the slope (31 degrees).
Now, let's calculate the vertical distance:
h = 117 * sin(31)
h ≈ 61.26 meters
Next, we can calculate the potential energy at the top of the slope:
P.E. = (45 + 5) * 9.8 * 61.26
To find the speed at the bottom of the slope, we need to use conservation of energy. The total energy at the top is equal to the total energy at the bottom, which is the sum of potential energy and kinetic energy.
At the bottom, all the potential energy is converted into kinetic energy. The formula for kinetic energy is K.E. = 0.5 * m * v^2, where m is the total mass (the child and the sled), and v is the speed.
Now, let's set up the equation for conservation of energy:
P.E. at the top = K.E. at the bottom
(45 + 5) * 9.8 * 61.26 = 0.5 * (45 + 5) * v^2
Simplifying,
50 * 9.8 * 61.26 = 25 * v^2
Now, let's solve for v:
v^2 = (50 * 9.8 * 61.26) / 25
v ≈ 43.37 m/s
Therefore, the child's speed at the bottom of the slope is approximately 43.37 m/s.