A 70 kg bicyclist rides his 7.8 kg bicycle with a speed of 14 m/s. How much work must be done by the brakes to bring the bike and rider to a stop?
How far does the bicycle travel if it takes 3.5 s to come to rest?
KE= work to be done
distance= averagevelocity*time= 1/2 starting velocity*time
To find the work done by the brakes to bring the bike and rider to a stop, we need to calculate the change in kinetic energy.
First, let's find the initial kinetic energy (KE_i) of the bicycle and rider:
KE_i = 1/2 * mass * velocity^2
Given:
Mass of the rider (m_rider) = 70 kg
Mass of the bicycle (m_bicycle) = 7.8 kg
Speed (velocity) = 14 m/s
Substituting the values into the formula, we can find the initial kinetic energy (KE_i) of the bicycle and rider:
KE_i = 1/2 * (m_rider + m_bicycle) * velocity^2
Next, let's find the final kinetic energy (KE_f) when the bicycle and rider come to a stop. The final kinetic energy would be zero as the bicycle comes to rest.
Now, the work done by the brakes (W) to bring the bicycle to a stop is equal to the change in kinetic energy:
W = KE_f - KE_i = 0 - KE_i = -KE_i
To find the distance traveled by the bicycle when it takes 3.5 seconds to come to rest, we can use the formula for distance traveled (d) during uniform acceleration:
d = (1/2) * acceleration * time^2
Since we know the initial velocity (v_i) = 14 m/s and final velocity (v_f) = 0 m/s, we can calculate the acceleration:
v_f = v_i + acceleration * time
0 = 14 + acceleration * 3.5
acceleration = -14 / 3.5
Now, we can substitute the values into the formula to find the distance traveled (d):
d = (1/2) * acceleration * time^2
d = (1/2) * (-14 / 3.5) * (3.5)^2
Let's calculate the results step by step.
To calculate the work done by the brakes to bring the bike and rider to a stop, we need to use the equation:
Work (W) = Force (F) x Distance (d) x Cosine (θ),
where:
- Work is the amount of energy transferred to bring the bike and rider to a stop,
- Force is the net force acting on the bike and rider,
- Distance is the distance traveled by the bike and rider before coming to a stop, and
- Cosine (θ) is the angle between the force vector and the displacement vector.
In this case, the net force acting on the bike and rider is the force applied by the brakes. When braking, the force applied by the brakes is equal to the force of friction between the tires and the road. The force of friction can be calculated using the equation:
Force of friction (Ff) = coefficient of friction (μ) x normal force (Fn),
where:
- Coefficient of friction (μ) is a value that depends on the nature of the surfaces in contact, and
- Normal force (Fn) is the force exerted by the surface supporting the bike and rider, which is equal to their combined weight.
Let's start by calculating the force of friction:
Weight of the bike and rider (Wbr) = mass of bike and rider (mbr) x acceleration due to gravity (g).
Wbr = (70 kg + 7.8 kg) x 9.8 m/s²,
Wbr = 70 kg x 9.8 m/s² + 7.8 kg x 9.8 m/s²,
Wbr = 686 N + 76.44 N,
Wbr = 762.44 N.
Assuming a coefficient of friction of μ = 0.6, we can calculate the force of friction:
Ff = 0.6 x 762.44 N,
Ff = 457.47 N.
Now, we can calculate the work done by the brakes:
Work (W) = Ff x d x Cosine (θ).
Since the bike and rider come to a stop, the net force acting on them is in the opposite direction to their motion. Therefore, θ = 180°.
Work (W) = 457.47 N x d x Cosine (180°),
Work (W) = -457.47 N x d x (-1), [since Cosine (180°) = -1],
Work (W) = 457.47 N x d.
Now, let's calculate how far the bicycle travels to come to rest.
We know that:
- Initial velocity (u) = 14 m/s,
- Final velocity (v) = 0 m/s, [since the bike comes to rest],
- Acceleration (a) = ?,
- Time (t) = 3.5 s.
Using the equation:
v = u + a.t,
0 = 14 m/s + a x 3.5 s.
Rearranging the equation:
a x 3.5 s = -14 m/s, [since final velocity is zero],
a = -14 m/s ÷ 3.5 s,
a = -4 m/s².
Since acceleration is negative, it indicates deceleration.
To find the distance (d), we can use the equation:
d = u.t + 0.5.a.t².
d = 14 m/s x 3.5 s + 0.5 x (-4 m/s²) x (3.5 s)²,
d = 49 m + (-0.5) x 16.625 m,
d = 49 m - 8.3125 m,
d = 40.6875 m.
Therefore, the work done by the brakes to bring the bike and rider to a stop is 457.47 Joules, and the bicycle travels a distance of 40.6875 meters to come to rest.