A car and driver weighing 7740 N passes a
sign stating “Bridge Out 27.7 m Ahead.” She slams on the brakes, and the car decelerates at a constant rate of 12.3 m/s^2.
The acceleration of gravity is 9.8 m/s^2.
What is the magnitude of the work done
stopping the car if the car just stops in time to avoid diving into the water?
Answer in units of J.
Well, considering the car is avoiding diving into water, maybe it's trying to land a high dive trick instead? Either way, let's calculate the work done.
The work done can be calculated using the formula:
Work = Force x Distance
In this case, the force can be calculated using Newton's second law: F = m x a, where F is the force, m is the mass, and a is the acceleration.
First, we need to find the mass of the car. We can use the formula: Weight = mass x gravity. Rearranging the formula, mass = Weight / gravity.
Given that the weight of the car is 7740 N and the acceleration due to gravity is 9.8 m/s^2, we can calculate the mass of the car.
mass = 7740 N / 9.8 m/s^2
Now that we have the mass of the car, we can calculate the force using F = m x a.
force = m x a
Next, we need to find the distance over which the car decelerates, which is given as 27.7 m.
Finally, we can calculate the work done using the formula: Work = Force x Distance.
Sorry if the explanation took a detour, but now let's crunch some numbers and find the answer!
To find the magnitude of the work done to stop the car, we can use the work-energy principle. According to this principle, the work done on an object is equal to the change in its kinetic energy.
First, let's find the initial velocity of the car. We can use the formula:
v^2 = u^2 + 2as
where v is the final velocity (which is 0 m/s since the car stops), u is the initial velocity, a is the deceleration (negative since it opposes motion), and s is the displacement.
Rearranging the formula, we have:
u^2 = v^2 - 2as
u^2 = 0 - 2 * (-12.3 m/s^2) * 27.7 m
u^2 = 2 * 12.3 m/s^2 * 27.7 m
u^2 = 684.81 m^2/s^2
Taking the square root of both sides, we get:
u = sqrt(684.81 m^2/s^2)
u ≈ 26.17 m/s
Now, we can calculate the initial kinetic energy (E_ki) of the car using the formula:
E_ki = 0.5 * m * u^2
where m is the mass of the car.
To find the mass of the car, we can use the weight (W) and the acceleration due to gravity (g) using the formula:
W = m * g
Rearranging the formula, we get:
m = W / g
m = 7740 N / 9.8 m/s^2
m ≈ 791.84 kg
Now, we can substitute the values into the formula for initial kinetic energy:
E_ki = 0.5 * 791.84 kg * (26.17 m/s)^2
E_ki ≈ 213,277.5 J
Therefore, the magnitude of the work done to stop the car is approximately 213,277.5 Joules.
To find the magnitude of the work done in stopping the car, we can use the work-energy theorem. The work done is equal to the change in kinetic energy.
The initial kinetic energy of the car is given by KE_initial = (1/2)mv^2, where m is the mass of the car and v is its initial velocity.
The final velocity of the car is 0 m/s since it just stops in time to avoid diving into the water.
The change in kinetic energy is therefore given by ΔKE = KE_final - KE_initial = -KE_initial.
The work done is negative because the force applied by the brakes is opposite to the direction of motion.
The work done can be expressed as W = Fd, where F is the force applied by the brakes and d is the distance over which the force is applied.
Since the force applied is equal to the net force, which in this case is equal to the product of the mass of the car and its deceleration, F = -ma.
The distance over which the force is applied is given as 27.7 m.
Substituting the given values, we have F = -ma = -m(-12.3 m/s^2) = 12.3m.
Now, the work done can be written as W = 12.3md.
We know the weight of the car and driver, which is given as 7740 N. The weight is equal to the mass of the car and driver multiplied by the acceleration due to gravity, W = mg. Therefore, mg = 7740 N.
Solving for the mass, m = 7740 N / 9.8 m/s^2.
Now, we can substitute the values of m and d into the equation for work to calculate the magnitude of the work done.
W = 12.3(7740 N / 9.8 m/s^2)(27.7 m)
Evaluating this expression will give us the magnitude of the work done when the car stops in time to avoid diving into the water.