The parachute on a race car of weight 8,810 N opens at the end of a quarter-mile run when the car is traveling at 34 m/s. What total retarding force must be supplied by the parachute to stop the car in a distance of 1,100 m?
V^2 = Vo^2 + 2a*d.
a = (V^2-Vo^2)/2d.
a = (0-(34)^2)/2200 = -0.525 m/s^2.
m*g = 8810N.
m = 8810/9.81 = 899 kg = Mass of car.
F = m*a = 899 * (-525) = -472 N.
To calculate the total retarding force required to stop the car, we need to determine the net force acting on the car.
First, let's calculate the initial kinetic energy of the car at the beginning of the quarter-mile run. We can use the formula:
KE = (1/2) * m * v^2
where:
KE is the kinetic energy,
m is the mass of the car,
v is the velocity of the car.
Given:
Weight of the car (W) = 8,810 N
g (acceleration due to gravity) = 9.8 m/s^2
Velocity (v) = 34 m/s
The weight of the car is related to the mass by the equation:
W = m * g
Rearranging the equation to solve for the mass:
m = W / g
Substituting the given values:
m = 8,810 N / 9.8 m/s^2
m ≈ 898.98 kg
Now we can calculate the initial kinetic energy:
KE = (1/2) * 898.98 kg * (34 m/s)^2
KE ≈ 514,891.876 J
Next, we need to calculate the final kinetic energy of the car when it comes to a stop. The final kinetic energy will be zero since the car comes to a stop.
Using the formula for kinetic energy, we have:
KE = (1/2) * m * v^2
Setting KE = 0 and solving for v:
0 = (1/2) * 898.98 kg * v^2
v^2 = 0
Therefore, the final velocity (v) of the car is 0 m/s.
Now, let's calculate the work done by the retarding force of the parachute to bring the car to a stop. The work done is equal to the change in kinetic energy:
Work = KE_final - KE_initial
Substituting the values:
Work = 0 - 514,891.876 J
Work ≈ -514,891.876 J
Finally, the total retarding force (F) can be calculated using the work-energy principle:
Work = F * d
where:
F is the force,
d is the distance over which the force is applied.
Given:
Work = -514,891.876 J
Distance (d) = 1,100 m
Solving for F:
-514,891.876 J = F * 1,100 m
F = -514,891.876 J / 1,100 m
Therefore, the total retarding force required to stop the car is approximately -468.083 N (assuming the negative sign indicates that the force is opposite to the direction of motion).
To find the total retarding force exerted by the parachute, we'll use Newton's second law of motion:
Force = Mass x Acceleration
First, let's find the mass of the race car. We'll use the weight and the acceleration due to gravity (g), which is approximately 9.8 m/s²:
Weight = Mass x g
8,810 N = Mass x 9.8 m/s²
Rearranging the equation to solve for mass:
Mass = Weight / g
Mass = 8,810 N / 9.8 m/s²
Mass ≈ 898 kg
Now, let's find the acceleration of the race car using the initial velocity (v₁), final velocity (v₂), and distance (d) covered:
v₁ = 34 m/s
v₂ = 0 m/s (since the car needs to stop)
d = 1,100 m
We'll use the formula of motion:
v₂² = v₁² + 2a(d - d₁)
where d₁ is the distance covered during the quarter-mile run (or 400 m). Rearranging this equation to solve for acceleration:
a = (v₂² - v₁²) / (2(d - d₁))
Substituting the values:
a = (0² - 34²) / (2(1,100 - 400))
a = (-1,156) / (2(700))
a ≈ -0.8286 m/s²
Since the car needs to stop, the acceleration is negative.
Finally, we can find the total retarding force:
Force = Mass x Acceleration
Force = 898 kg x (-0.8286 m/s²)
Force ≈ -742.57 N
The total retarding force supplied by the parachute must be approximately 742.57 N (rounded to three decimal places).