A driver of a 7400 N car passes a sign stating "Bridge Out 30 Meters Ahead." She slams on the brakes, coming to a stop in 10 s. How much work must be done by the brakes on the car if it is to stop just in time? Neglect the weight of the driver, and assume that the negative acceleration of the car caused by the braking is constant

To find the work done by the brakes on the car, we need to calculate the force exerted by the brakes and multiply it by the distance over which the braking force acts.

First, let's calculate the deceleration (negative acceleration) of the car. We can use the formula:

acceleration = change in velocity / time

Since the car comes to a stop, the change in velocity is equal to the initial velocity (v₀) of the car. Assuming the car was initially traveling at a constant speed, the change in velocity is given by:

change in velocity = v - v₀,

where v is the final velocity (which is 0 in this case).

Given that the car stops in 10 seconds, the acceleration becomes:

acceleration = (0 - v₀) / 10

Next, we can calculate the force exerted by the brakes using Newton's second law:

force = mass * acceleration.

Here, the mass is given by the weight divided by the acceleration due to gravity:

mass = weight / acceleration due to gravity,

where the weight is given as 7400 N and the acceleration due to gravity is approximately 9.8 m/s².

Finally, we can find the work done by the brakes by multiplying the force by the distance over which the braking force acts. The distance given is 30 meters.

work = force * distance.

Let's calculate the values step-by-step:

1. Calculate the acceleration:
acceleration = (0 - v₀) / 10.

2. Calculate the mass:
mass = weight / acceleration due to gravity.

3. Calculate the force exerted by the brakes:
force = mass * acceleration.

4. Calculate the work done by the brakes:
work = force * distance.

To determine the work done by the brakes on the car, we first need to calculate the force exerted by the brakes.

Using Newton's second law of motion, which states that force is equal to mass multiplied by acceleration (F = m * a), we can rearrange the equation to find the acceleration:

F = m * a
a = F / m

The force required to bring the car to a stop is the net force acting on the car, which is the difference between the frictional force provided by the brakes and the gravitational force acting on the car:

F_net = F_brakes - F_gravity

Since the car is coming to a stop, its acceleration is in the opposite direction of its velocity. Therefore, the acceleration is negative (-a).

Now, let's find the gravitational force acting on the car. The weight of the car is given as 7400 N. The gravitational force is equal to the weight, so F_gravity = 7400 N.

Next, we can calculate the acceleration of the car:

a = F_net / m
a = (F_brakes - F_gravity) / m

The negative sign indicates that the car is decelerating.

Once we have the acceleration, we can use the kinematic equation to find the initial velocity of the car. The equation for calculating the final velocity (vf) given the initial velocity (vi), acceleration (a), and time (t) is:

vf = vi + a * t

Since the final velocity when the car comes to a stop is zero, we can rearrange the equation to solve for the initial velocity:

0 = vi + a * t
vi = -a * t

Now, we have all the necessary information to calculate the required work.

The work done during braking is given by the formula:

Work = Force * Distance

To find the distance, we need to calculate the initial velocity of the car. The distance is equal to the initial velocity multiplied by the time to stop:

Distance = vi * t

Substituting the value of vi from earlier, we get:

Distance = (-a * t) * t

Finally, we can calculate the work done by the brakes:

Work = (F_brakes - F_gravity) * Distance

Substituting the values for F_gravity, a, and Distance, we can determine the work required.