3. A skater with an initial speed of 7.6 m/s is gliding across the ice. The coefficient of friction between the ice and the skae blades is μk=0.1.
a. Find acceleration of the skater
b. How foar will the skater travel before coming to rest
mass m
weight = m g = 9.81 m
friction force = 0.1 * weight
= 0.981 m
That is the entire horizontal force
so
F = m a
0.981 m = m a
a = 0.981
-------------------
how long to stop?
v = Vi - 0.981 t
0 = 7.6 - 0.981 t
t = 7.75 seconds to grind to a halt
d = Vi t - (1/2)(0.981) t^2
d = 7.6 (7.75) - .49 (7.75)^2
= 58.8 - 29.4
= 29.4 meters
a. To find the acceleration of the skater, we can use the equation of motion:
Acceleration = μk * g
where μk is the coefficient of friction and g is the acceleration due to gravity.
Given that μk = 0.1 and g ≈ 9.8 m/s², we can substitute these values into the equation:
Acceleration = 0.1 * 9.8 = 0.98 m/s²
Therefore, the acceleration of the skater is 0.98 m/s².
b. To find how far the skater will travel before coming to rest, we can use the equation of motion:
v² = u² + 2as
where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the distance traveled.
In this case, 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/s² (since it is decelerating). We can substitute these values into the equation:
0 = (7.6)² + 2 * (-0.98) * s
Simplifying the equation:
0 = 57.76 - 1.96s
1.96s = 57.76
s = 57.76 / 1.96
s ≈ 29.51 meters
Therefore, the skater will travel approximately 29.51 meters before coming to rest.
To find the acceleration of the skater, we can use the equation:
\[ \text{Net force} = \text{mass} \times \text{acceleration} \]
The net force acting on the skater is the force of friction. The force of friction can be calculated using the equation:
\[ \text{Force of friction} = \text{coefficient of friction} \times \text{normal force} \]
The normal force is equal to the weight of the skater, which can be calculated using the equation:
\[ \text{Normal force} = \text{mass} \times \text{gravity} \]
where gravity is approximately 9.8 m/s².
Let's calculate the acceleration:
a.
Given:
Initial speed (u) = 7.6 m/s
Coefficient of friction (μk) = 0.1
First, we need to calculate the normal force:
\[ \text{Normal force} = \text{mass} \times \text{gravity} \]
Substituting the given values of the coefficient of friction and acceleration due to gravity:
\[ \text{Normal force} = \text{mass} \times 9.8 \]
Next, we can calculate the force of friction:
\[ \text{Force of friction} = \text{coefficient of friction} \times \text{normal force} \]
Substituting the given value of the coefficient of friction and the previously calculated value of the normal force:
\[ \text{Force of friction} = 0.1 \times \left(\text{mass} \times 9.8\right) \]
Now, we can calculate the acceleration using the equation:
\[ \text{Net force} = \text{mass} \times \text{acceleration} \]
Since the net force is equal to the force of friction, we have:
\[ 0.1 \times \left(\text{mass} \times 9.8\right) = \text{mass} \times \text{acceleration} \]
We can simplify this equation to solve for acceleration:
\[ \text{acceleration} = 0.1 \times 9.8 \]
Substituting the value of gravity:
\[ \text{acceleration} = 0.1 \times 9.8 \]
Calculating the value of acceleration:
\[ \text{acceleration} = 0.98 \, \text{m/s}^2 \]
b.
To find how far the skater will travel before coming to rest, we can use the equation:
\[ \text{Distance} = \frac{{\text{Initial velocity}^2}}{{2 \times \text{acceleration}}} \]
Substituting the known values:
\[ \text{Distance} = \frac{{7.6^2}}{{2 \times 0.98}} \]
Calculating the distance:
\[ \text{Distance} = 37.959 \, \text{m} \]
Therefore, the skater will travel approximately 37.959 meters before coming to rest.