A car and driver weighing 7750 N passes a sign stating “Bridge Out 28.5 m Ahead.” She slams on the brakes, and the car decelerates at a constant rate of 10.6 m/s2 .
The acceleration of gravity is 9.8 m/s2 .
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
Using Newton's second law F=ma
to find the mass, then again the same law to find the frictional force.
Finally use Work done = F.D
Units are in kg, m/s² and J.
Mass m = 7750/9.8 = 790.82 kg
Frictional force = ma
= 790.82 kg * 10.6 m/s²
= 8382.7 N
Work done (assuming uniform frictional foce)
= F.D
= 8382.7 N * 28.5 m
= 238906 J
= 238.9 kJ
Well, let's calculate the work done, but before that, can we take a moment to appreciate the irony? A car diving into the water would definitely make for a great splash! Anyway, back to the question at hand.
The work done is equal to the force applied multiplied by the distance over which it acts. In this case, the force is the product of mass and acceleration (F = m * a). So, first, let's calculate the mass of the car.
The weight of an object is given by W = m * g, where g is the acceleration due to gravity. Rearranging the equation, we have: m = W / g.
Substituting the given values, we get: m = 7750 N / 9.8 m/s^2 = 790.82 kg (approximately).
Now, let's calculate the work done. The distance over which the force acts is 28.5 m.
Using the work formula, W = F * d, substituting the values, we get: W = (m * a) * d = (790.82 kg * -10.6 m/s^2) * 28.5 m.
Calculating the expression, we find that the magnitude of the work done is approximately equal to -231429 J.
So, the magnitude of the work done to stop the car just in time to avoid diving into the water is approximately 231429 J.
Remember to cherish the irony and keep those dives for the pool, not the bridge!
To find the work done in stopping the car, we can use the work-energy principle. The work done on an object is equal to its change in kinetic energy.
The initial kinetic energy of the car can be calculated using the formula:
KE_initial = (1/2) * m * v^2
Where m is the mass of the car and v is the initial velocity.
Given that the car and driver weigh 7750 N, we can find the mass of the car (m) by dividing the weight by the acceleration due to gravity (g = 9.8 m/s^2):
m = 7750 N / 9.8 m/s^2
Substituting the values, we get:
m = 791.8367 kg (approximately)
The initial velocity (v) of the car is not given in the question. However, we do not need it to find the magnitude of the work done. We can use a different approach.
The final velocity of the car is 0 m/s since it just stops in time to avoid diving into the water. Therefore, the change in velocity (Δv) is:
Δv = 0 - v = -v
Now, we can use the definition of acceleration to relate the change in velocity with the deceleration:
a = Δv / t
Where a is the deceleration and t is the time taken to stop.
Rearranging the equation, we get:
t = -Δv / a
Substituting the values, we get:
t = -(-v) / 10.6 m/s^2 = v / 10.6 m/s^2
Now, we can calculate the work done using the equation:
Work_done = KE_final - KE_initial
Since the final kinetic energy (KE_final) is zero, it simplifies to:
Work_done = -KE_initial
Substituting the values, we get:
Work_done = -(1/2) * m * v^2
Work_done = -(1/2) * 791.8367 kg * (v^2)
As we can see, the magnitude of work done stopping the car depends on the square of the initial velocity (v). Since the initial velocity is not given in the question, we cannot determine the exact value of the work done without that information.
To find the magnitude of the work done in stopping the car, we can use the formula for work: work = force × distance × cosine of the angle between the force and displacement.
In this case, the force acting on the car is the force of friction, opposing its motion. The force of friction can be calculated using the equation: force = mass × acceleration.
First, let's find the mass of the car and driver. We are given the weight of the car and driver, which is the force due to gravity acting on them. The weight of an object is given by the equation: weight = mass × acceleration due to gravity.
Thus, we can rearrange the equation to solve for mass:
mass = weight ÷ acceleration due to gravity.
Given that the weight is 7750 N and the acceleration due to gravity is 9.8 m/s^2, we can calculate the mass of the car and driver.
mass = 7750 N ÷ 9.8 m/s^2.
mass ≈ 790.82 kg.
Next, we need to determine the distance the car travels in order to stop. The distance can be found using the equation of motion:
velocity^2 = initial velocity^2 + 2 × acceleration × distance.
In this case, the car starts from rest (initial velocity = 0 m/s), decelerates at a constant rate of -10.6 m/s^2 (negative because it's opposing motion), and we need to find the distance.
By rearranging the equation, we have:
distance = (velocity^2 - initial velocity^2) ÷ (2 × acceleration).
Since the final velocity is zero (the car stops), the equation simplifies to:
distance = - initial velocity^2 ÷ (2 × acceleration).
Given that the initial velocity is zero and the acceleration is -10.6 m/s^2, we can now calculate the distance:
distance = 0 - 0 ÷ (2 × (-10.6 m/s^2)).
distance = 0 m.
Since the distance is zero, we can conclude that the car stops exactly in time to avoid diving into the water.
Now we can calculate the work done.
work = force × distance × cosine of the angle between the force and displacement.
In this case, the angle between the force of friction and the displacement is 0 degrees, as the force and displacement are in the same direction.
Therefore, the cosine of 0 degrees is 1.
work = force × distance × 1.
Now, we substitute the force and distance values into the equation:
work = (mass × acceleration) × distance × 1.
work = (790.82 kg × -10.6 m/s^2) × 0 m × 1.
work = 0 J.
Therefore, the magnitude of the work done in stopping the car is 0 J.