Question

A sled has an initial velocity after coming off the edge of a small hill. Once the sled hits the bottom of the hill, it starts to slow down until it comes to rest. If the initial speed of the sled was 12 m/s and the coefficient of kinetic

friction between the snow and the sled is 0.05, what is the acceleration of the sled as it comes to a stop?
A box slides down a 30 degree ramp with an acceleration of 1.20 m/s². Determine the coefficient of kinetic friction between the box and the ramp.

Answer 1

For the sled, the acceleration can be found using the equation:

`\[a = g \cdot \mu_k\]`
where
`\(a\)` is the acceleration,
`\(g\)` is the acceleration due to gravity (approximately 9.8 m/s²), and
`\(\mu_k\)` is the coefficient of kinetic friction.




For the box on the ramp, the acceleration can be determined using the equation:


`\[a = g \cdot (\sin(\theta) - \mu_k \cdot \cos(\theta))\]`

where
`\(\theta\)` is the angle of the ramp.

Let's calculate:

1. For the sled:

`\[a_{\text{sled}} = 9.8 \, \text{m/s}^2 \cdot 0.05 = 0.49 \, \text{m/s}^2\]`

2. For the box on the ramp:

`\[1.20 \, \text{m/s}^2 = 9.8 \, \text{m/s}^2 \cdot (\sin(30^\circ) - \mu_k \cdot \cos(30^\circ))\]`

Solving for
`\(\mu_k\)`:


`\[\mu_k = (\sin(30^\circ) - (1.20)/(9.8))/(\cos(30^\circ))\]`
After calculating,
`\(\mu_k \approx 0.168\)`.

Answer 2

The acceleration of the sled is 0.49 m/s² due to kinetic friction. To determine the coefficient of kinetic friction between the box and the ramp, an equation relating the acceleration, gravitational force components, and friction must be used.

Step-by-step explanation:

Calculating the Acceleration Due to Friction

To find the acceleration of the sled as it comes to a stop due to kinetic friction, we can use the formula a = f_k / m, where a is the acceleration, f_k is the kinetic frictional force, and m is the mass of the sled. The kinetic frictional force can be calculated using the equation f_k = μ_k × N, where μ_k is the coefficient of kinetic friction and N is the normal force. Given that μ_k = 0.05 and assuming the normal force is equal to the weight of the sled (which is the mass times the acceleration due to gravity, g), the frictional force is f_k = 0.05 × m × g. Thus, a = (0.05 × m × g) / m, which simplifies to a = 0.05 × g. Taking g to be approximately 9.8 m/s², the acceleration due to friction is a = 0.05 × 9.8 m/s², or a = 0.49 m/s², in the direction opposite to the sled's motion.

Coefficient of Kinetic Friction on a Ramp

To determine the coefficient of kinetic friction between the box and the ramp, we can use the equation μ_k = (a - g × sinθ) / (g × cosθ), where a is the acceleration of the box, g is the acceleration due to gravity, and θ is the angle of the ramp with the horizontal. Here, a is given as 1.20 m/s² and θ is 30 degrees. Substituting the known values, the coefficient of kinetic friction can be calculated.