3. Johnny, of mass 65 kg, and Lucy, of mass 45 kg, are facing each other on roller blades. The coefficient of kinetic friction between the roller blades and concrete surface is 0.20. When Johnny pushes Lucy from rest he applies a force for 1.0 s. Lucy then slows down to a stop in another 8.0 s. Calculate:
a.The applied force exerted by Johnny on Lucy.
b.How long it takes Johnny to come to rest
a. To calculate the applied force exerted by Johnny on Lucy, we can use Newton's second law of motion, which states that force (F) is equal to mass (m) multiplied by acceleration (a):
F = m * a
First, we need to find the acceleration of Lucy when Johnny pushes her. We can use the equation:
acceleration = change in velocity / time
Since Lucy starts from rest, her initial velocity (u) is 0. The final velocity (v) can be calculated as:
v = u + a * t
where t is the time for which Johnny applies the force (1.0 s).
Substituting the values, we get:
v = 0 + a * 1
We know that the force of friction (f) between Lucy and the ground can be calculated using the equation:
f = coefficient of friction * normal force
The normal force (N) can be found by multiplying Lucy's mass (m) by the acceleration due to gravity (g).
N = m * g
where g is approximately 9.8 m/s^2.
The force applied by Johnny can be calculated by subtracting the force of friction from the force required to accelerate Lucy:
F = m * a + f
Substituting the values, we get:
F = m * a + (coefficient of friction * m * g)
Now, let's plug in the numbers:
m (mass of Lucy) = 45 kg
m (mass of Johnny) = 65 kg
coefficient of friction (μ) = 0.20
g (acceleration due to gravity) = 9.8 m/s^2
Calculating the normal force:
N = m * g
N = 45 kg * 9.8 m/s^2
N = 441 N
Calculating the force of friction:
f = coefficient of friction * normal force
f = 0.20 * 441 N
f = 88.2 N
Now we can calculate the applied force exerted by Johnny on Lucy:
F = m * a + f
F = 45 kg * a + 88.2 N
Note that we need the acceleration 'a' to calculate the force 'F'. Unfortunately, we do not have enough information to calculate the acceleration from the given data. Please provide additional information, such as the distance traveled by Lucy during the 8.0 s.
b. Similarly, we need more information to calculate the time it takes for Johnny to come to rest.
a. To calculate the applied force exerted by Johnny on Lucy, we need to use Newton's second law of motion. The equation for the force (F) is given by:
F = m * a
Where:
F is the force applied
m is the mass of Lucy
a is the acceleration of Lucy
We can find the acceleration using the equation:
a = Δv / Δt
Where:
Δv is the change in velocity of Lucy
Δt is the time interval Johnny applies the force
Since Lucy goes from rest to a stop, her change in velocity is equal to her initial velocity. Therefore:
Δv = v_initial - 0 = v_initial
Substituting these values into the equation, we have:
a = v_initial / Δt
Next, we need to determine the net force acting on Lucy, which is given by:
F_net = μ * m * g
Where:
F_net is the net force (opposite to the applied force) acting on Lucy
μ is the coefficient of kinetic friction between the roller blades and the concrete surface
m is the mass of Lucy
g is the acceleration due to gravity (approximately 9.8 m/s^2)
Since the applied force by Johnny and the net force are in opposite directions, we can write:
F_applied = F_net
Substituting the values, we have:
m * a = μ * m * g
Now, we can solve for the applied force (F_applied):
F_applied = m * a
F_applied = m * (v_initial / Δt)
F_applied = m * v_initial / Δt
Substituting the given values, we have:
F_applied = (45 kg) * v_initial / (1.0 s)
b. To calculate how long it takes Johnny to come to rest, we need to use the equation for the time interval (Δt):
Δt = Δv / a
Since we know that the acceleration will be equal to the net force acting on Johnny divided by his mass, and the net force is equal to the applied force from Lucy divided by his mass, we can write:
Δt = Δv / (F_applied / m)
Substituting the values, we have:
Δt = v_initial / (F_applied / m)
Δt = v_initial / ((45 kg * v_initial / (1.0 s)) / 65 kg)
Simplifying, we have:
Δt = v_initial / ((45 kg * v_initial / 1.0 s) / 65 kg)
Now, we have the equations to find both the applied force (F_applied) and the time interval (Δt). We just need to know the value of the initial velocity (v_initial). Could you please provide the initial velocity of Lucy?