You pull your little sister across a flat snowy field on a sled. Your sister plus the sled have a mass of 39 kg. The rope is at an angle of 39 degrees to the ground. As you pull with a force of 31 N, the sled travels a distance of 47 m.
There is no question here
To solve this problem, we can use Newton's second law of motion, which states that the sum of the forces acting on an object is equal to the product of its mass and acceleration.
First, let's break down the force you're applying into its horizontal and vertical components. The force you're applying can be decomposed into two directions: one parallel to the ground (the horizontal component) and one perpendicular to the ground (the vertical component).
The vertical component of the force does not affect the sled's motion since it's perpendicular to the direction of motion. Therefore, we can ignore it for now and focus only on the horizontal component.
To find the horizontal component of the force, we need to determine the angle between the force and the horizontal direction. In this case, the angle is given as 39 degrees.
The horizontal component of the force (F_hor) is calculated by multiplying the magnitude of the force (31 N) by the cosine of the angle (39 degrees):
F_hor = 31 N * cos(39 degrees)
Next, we need to calculate the acceleration of the sled. We know that force equals mass times acceleration (F = m * a). In this case, the mass of the sled and your sister combined is 39 kg. Rearranging this equation, we can solve for acceleration:
a = F_hor / m
Now that we have the acceleration, we can use it to find the time it takes for the sled to travel the given distance of 47 m using the third equation of motion:
s = ut + 0.5at^2
In this equation, s represents the distance, u is the initial velocity (which we assume to be zero since the sled starts from rest), t is the time, and a is the acceleration we just calculated.
By rearranging the equation, we can solve for time:
t = sqrt(2s / a)
Substituting the values we have, we can find the time it takes for the sled to travel 47 m.
After finding the time, we can calculate the final velocity of the sled using the equation:
v = u + at
Since the sled starts from rest (u = 0), the final velocity is the same as the velocity calculated using this equation.
Now, let's plug in the given values and calculate the results step by step.