Maryam is riding her bicycle on a path when she comes around a corner and sees that a fallen
tree is blocking the way. If the coefficient of friction between her bicycle's tires and the gravel
path is 0.36 and she is traveling at 22.0 km/h, how much stopping distance will she require?
Maryam and her bicycle, together, have a mass of 100 kg.
5.3
To find the stopping distance, we need to determine the net force acting on Maryam and her bicycle while she is braking. The net force can be calculated using Newton's second law: Fnet = m * a, where Fnet is the net force, m is the mass, and a is the acceleration.
First, we need to find the initial velocity of Maryam's bicycle in meters per second (m/s). Given that she is traveling at 22.0 km/h, we can convert it to m/s by multiplying it by 1000/3600:
22.0 km/h * (1000 m/3600 s) = 6.11 m/s
Next, we need to find the final velocity of the bicycle when it comes to a complete stop. Since Maryam is stopping, the final velocity will be 0 m/s.
Now, we can calculate the change in velocity (Δv) using the equations:
Δv = final velocity - initial velocity
Δv = 0 m/s - 6.11 m/s = -6.11 m/s
To find the deceleration (negative acceleration) of the bicycle, we can use the following equation: a = Δv / t, where a is the acceleration and t is the time taken to stop.
Since we don't know the time taken to stop, let's calculate it using the formula for distance traveled during uniform acceleration:
Δx = (v_f^2 - v_i^2) / (2*a)
Where Δx is the stopping distance (the distance travelled while braking), v_f is the final velocity (0 m/s), and v_i is the initial velocity (6.11 m/s).
Let's plug in the values:
Δx = (0^2 - 6.11^2) / (2*a)
Δx = (-6.11^2) / (2*a)
We need to solve for a. To do that, we can use the frictional force equation:
F_f = μ * m * g
Where F_f is the force of friction, μ is the coefficient of friction (0.36), m is the mass (100 kg), and g is the acceleration due to gravity (approximately 9.8 m/s^2).
F_f = μ * m * g
F_f = 0.36 * 100 kg * 9.8 m/s^2
F_f = 352.8 N
Since the force of friction is equal to the net force (F_net), we can equate it to m * a:
m * a = F_f
100 kg * a = 352.8 N
Now we can solve for a:
a = 352.8 N / 100 kg
a = 3.528 m/s^2
Plugging the value of a back into the equation for stopping distance:
Δx = (-6.11^2) / (2 * 3.528 m/s^2)
Δx = 11.39 m
Therefore, Maryam will require a stopping distance of approximately 11.39 meters.
To calculate the stopping distance required, we need to consider the forces acting on the bicycle and use the laws of motion. The force of friction is the force that opposes the motion of the bicycle and brings it to a stop.
Here's how we can approach this problem:
Step 1: Convert Maryam's speed from km/h to m/s.
Given: Speed = 22.0 km/h
To convert km/h to m/s, divide the speed by 3.6:
22.0 km/h ÷ 3.6 = 6.11 m/s (rounded to two decimal places)
Step 2: Calculate the force of friction.
The force of friction can be determined using the equation: force of friction = coefficient of friction × normal force.
Given: coefficient of friction (μ) = 0.36
The normal force can be calculated using the formula: normal force = mass × gravity.
Given: mass (m) = 100 kg
gravity (g) = 9.8 m/s² (approximate value)
normal force = 100 kg × 9.8 m/s² = 980 N
force of friction = 0.36 × 980 N = 352.8 N (rounded to one decimal place)
Step 3: Calculate the stopping distance.
The force of friction acts as a decelerating force. We can use the equation: force of friction = mass × acceleration.
Rearranging the equation, we get: acceleration = force of friction ÷ mass.
acceleration = 352.8 N ÷ 100 kg = 3.528 m/s² (rounded to three decimal places)
Using the equation for motion, v² = u² + 2as, where:
v = final velocity (0 m/s when stopping)
u = initial velocity (6.11 m/s)
a = acceleration (3.528 m/s²)
s = stopping distance (what we are trying to find)
Substituting the known values into the equation and solving for s:
0² = (6.11 m/s)² + 2(3.528 m/s²)s
0 = 37.4521 m²/s² + 7.056s
Rearranging the equation to solve for s:
7.056s = -37.4521 m²/s²
s = -37.4521 m²/s² ÷ 7.056 ≈ -5.31 m²/s²
Since the stopping distance cannot be negative, we discard the negative value.
The stopping distance required is approximately 5.31 meters.
Therefore, Maryam will require a stopping distance of approximately 5.31 meters to come to a complete stop given the provided conditions.