A hockey puck slides with an initial speed of 43.2 m/s on a large frozen lake. If the coefficient of kinetic friction between the puck and the ice is 0.036, what is the speed of the puck after 10.4 s?
To find the speed of the hockey puck after 10.4 seconds, we need to calculate the acceleration of the puck first. Then, we can use the acceleration to find the final velocity.
1. Calculate the acceleration:
The net force acting on the hockey puck is the force of kinetic friction. The formula for the force of kinetic friction is given by:
F_friction = μ * F_normal
Where:
F_friction = force of kinetic friction
μ = coefficient of kinetic friction
F_normal = normal force
In this case, since the puck is on a horizontal surface, the normal force is equal to the weight of the puck. The formula for weight is given by:
F_weight = m * g
Where:
F_weight = weight of the puck
m = mass of the puck
g = acceleration due to gravity (approximately 9.8 m/s²)
The force of kinetic friction can be written as:
F_friction = μ * F_weight
The acceleration can be calculated using Newton's second law, which states that:
Net force = mass * acceleration
So, we can rewrite the equation as:
F_friction = m * a
Substituting the equation for force of friction, we get:
μ * F_weight = m * a
Simplifying, we find:
a = (μ * F_weight) / m
2. Calculate the final velocity:
We can use the equation of motion to calculate the final velocity. The equation is:
v_final = v_initial + (a * t)
Where:
v_final = final velocity
v_initial = initial velocity (given as 43.2 m/s)
a = acceleration (calculated in step 1)
t = time (given as 10.4 s)
Now, let's substitute the values into the equations to find the answer.
First, calculate the acceleration:
F_weight = m * g
F_friction = μ * F_weight
μ * F_weight = m * a
a = (μ * F_weight) / m
Calculate the weight of the puck:
F_weight = m * g
Substitute the weight into the equation for friction:
F_friction = μ * F_weight
Calculate the acceleration using the given values:
a = (μ * F_weight) / m
Then, calculate the final velocity:
v_final = v_initial + (a * t)
Substitute the known values into the equation to find the final velocity.