6. Drag-race tires in contact with an asphalt surface have a very high coefficient of static friction. Assuming a constant acceleration and no slipping of tires, estimate the coefficient of static friction needed for a drag racer to cover 2.1 km in 8 s, starting from rest.
Coefficient of static friction =
1.3
To estimate the coefficient of static friction needed for a drag racer to cover a given distance in a given time, we can use the kinematic equation:
d = 0.5 * a * t^2
where:
- d is the distance covered
- a is the acceleration
- t is the time taken
In this case, we have:
- d = 2.1 km = 2100 m (converting km to m)
- t = 8 s
The drag racer starts from rest, so its initial velocity (u) is 0.
We can rearrange the equation to solve for acceleration (a):
a = 2d / t^2
Substituting the given values, we get:
a = 2 * 2100 m / (8 s)^2
= 2 * 2100 m / 64 s^2
≈ 65.625 m/s^2
Now, we need to find the coefficient of static friction (μ) that is required to achieve this acceleration.
The formula for frictional force (f) is given by:
f = μ * N
Where:
- f is the frictional force
- μ is the coefficient of static friction
- N is the normal force
In this case, the normal force is equal to the weight of the drag racer (mass * gravity), but since we are assuming no slipping of tires, the frictional force is equal to the force of acceleration. So we can rewrite the equation as:
f = m * a
where:
- m is the mass of the drag racer
Since the frictional force is given by the mass times the acceleration, μ can be calculated using the rearranged equation:
μ = f / N
= (m * a) / (m * g)
= a / g
where:
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
Now, we can substitute the values to find μ:
μ = 65.625 m/s^2 / 9.8 m/s^2
≈ 6.7
Therefore, the coefficient of static friction needed for a drag racer to cover 2.1 km in 8 s, starting from rest, is approximately 6.7.