A skater with an initial speed of 7.6 m/s is gliding across the ice. Air resistance is negligible. The coefficient of kinetic friction between the ice and the blades is 0.1. a) Find the acceleration caused by kinetic friction and b) how far will the skater travel before coming to rest?
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To find the acceleration caused by kinetic friction, we can use the formula:
Acceleration = coefficient of kinetic friction * acceleration due to gravity
In this case, the coefficient of kinetic friction is given as 0.1, and the acceleration due to gravity is approximately 9.8 m/s^2.
a) Therefore, the acceleration caused by kinetic friction is:
Acceleration = 0.1 * 9.8 m/s^2
Acceleration = 0.98 m/s^2
b) To find how far the skater will travel before coming to rest, we can use the equation of motion:
Final velocity^2 = Initial velocity^2 + 2 * acceleration * distance
In this problem, the final velocity is 0 m/s (since the skater comes to rest), the initial velocity is 7.6 m/s, and the acceleration is -0.98 m/s^2 (since it acts opposite to the direction of motion).
Therefore, the equation becomes:
(0 m/s)^2 = (7.6 m/s)^2 + 2 * (-0.98 m/s^2) * distance
Simplifying this equation:
0 = 57.76 m^2/s^2 - 1.96 m^2/s^2 * distance
Rearranging the equation:
Distance = 57.76 m^2/s^2 / 1.96 m^2/s^2
Distance = 29.5 m
Therefore, the skater will travel approximately 29.5 meters before coming to rest.
To find the answers to both parts of the question, we will first need to use the formula for kinetic friction:
Frictional force (Ff) = Coefficient of kinetic friction (μk) × Normal force (N)
where:
- Ff is the force of kinetic friction,
- μk is the coefficient of kinetic friction, and
- N is the normal force.
For a skater gliding across the ice, since air resistance is negligible, the only force acting against the motion is kinetic friction. The normal force is equal to the skater's weight (mg), where m is the mass of the skater and g is the acceleration due to gravity (9.8 m/s^2).
a) Find the acceleration caused by kinetic friction:
The force of kinetic friction (Ff) is given by Ff = μk × N.
Since N = mg, we can substitute it into the formula to get Ff = μk × mg.
The force of friction also equals the product of the mass and acceleration (Ff = m × a), so we can equate the two:
μk × mg = m × a
Now, we can cancel out the mass (m) on both sides:
a = μk × g
Given that μk = 0.1 and g = 9.8 m/s^2, we can plug in the values to find the acceleration (a):
a = 0.1 × 9.8
a = 0.98 m/s^2
b) How far will the skater travel before coming to rest:
To determine how far the skater will travel before coming to rest, we can use the following equation:
v^2 = u^2 + 2aS
where:
- v is the final velocity (which is zero as the skater comes to rest),
- u is the initial velocity (7.6 m/s),
- a is the acceleration caused by kinetic friction (0.98 m/s^2), and
- S is the distance traveled.
Rearrange the equation to solve for S:
S = (v^2 - u^2) / (2a)
Since v = 0:
S = (0 - 7.6^2) / (2 × 0.98)
S = -58.24 / 1.96
S = -29.7 m^2/s^2
The distance traveled (S) cannot be negative, so we can take the magnitude of the answer:
S = 29.7 m
Therefore, the skater will travel approximately 29.7 meters before coming to rest.