A child on a sled reaches the bottom of a hill with a velocity of 10.0 and travels 23.0 along a horizontal straightaway to a stop.If the child and sled together have a mass of 58.0 , what is the magnitude of the average retarding force on the sled on the horizontal straightaway?
To find the magnitude of the average retarding force on the sled on the horizontal straightaway, you need to use Newton's second law, which states that force is equal to the mass of an object multiplied by its acceleration. In this case, the sled comes to a stop, so its final velocity is zero, while its initial velocity is 10.0 m/s. The distance traveled by the sled is given as 23.0 meters, and its mass is 58.0 kg.
First, calculate the acceleration of the sled using the equation:
acceleration = (final velocity - initial velocity)/time
Since the sled comes to a stop, its final velocity is zero. Rearranging the equation, we get:
acceleration = (0 - 10.0 m/s)/time
To find the time it takes for the sled to come to a stop, we need to use the equation:
distance = (initial velocity * time) + (0.5 * acceleration * time^2)
Rearranging the equation, we can solve for the time:
23.0 m = (10.0 m/s * time) + (0.5 * acceleration * time^2)
Now, substituting the acceleration value we got previously:
23.0 m = (10.0 m/s * time) + (0.5 * (0 - 10.0 m/s)/time * time^2)
Simplifying the equation gives:
23.0 m = (10.0 m/s * time) - (5.0 m/s^2 * time^2)
Rearranging to a quadratic equation:
5.0 m/s^2 * time^2 - 10.0 m/s * time + 23.0 m = 0
Solving this quadratic equation for time using the quadratic formula gives you two possible solutions, but only the positive time value is relevant since we are looking for the time it takes for the sled to stop. Let's say the positive time value is t.
Now, we can calculate the acceleration using the equation:
acceleration = (0 - 10.0 m/s)/t
Finally, we can find the magnitude of the average retarding force using Newton's second law:
force = mass * acceleration
Substituting the given values:
force = 58.0 kg * [(0 - 10.0 m/s)/t]
Simplify this equation using the value of t we found previously to get the magnitude of the average retarding force on the sled on the horizontal straightaway.