An advertisement claims that a particular automobile can "stop on a dime". What net force would actually be necessary to stop an automobile of mass 850kg traveling initially at a speed of 56.0km/h in a distance equal to the diameter of a dime, which is 1.8cm ?
a = (V^2-Vo^2)/2d
a = (0-(56)^2)/0.036 = -87,111 m/s^2
F = m*a = 850 * (-87,111)=-74,044,444 N.
To determine the net force required to stop the automobile, we need to use Newton's second law of motion, which states that the net force acting on an object is equal to its mass times its acceleration.
First, let's convert the initial speed from km/h to m/s. We can do this by dividing by 3.6 since there are 3.6 meters in one second:
56.0 km/h ÷ 3.6 = 15.6 m/s
Next, let's calculate the total distance the automobile needs to travel to come to a stop. The distance is given as equal to the diameter of a dime, which is 1.8 cm. Since the diameter is twice the radius, we can use 0.9 cm as the radius:
0.9 cm = 0.009 m
Therefore, the distance the automobile needs to travel is the circumference of a circle with a radius of 0.009 m, which can be calculated as:
2π(0.009 m) ≈ 0.0565 m
Now, we need to calculate the acceleration required to stop the automobile within this distance. The final velocity of the automobile is zero since it comes to a stop, so we can use the equation:
v^2 = u^2 + 2as
Where:
v = final velocity (0 m/s in this case)
u = initial velocity (15.6 m/s)
a = acceleration (which we want to find)
s = distance (0.0565 m)
Rearranging the equation to solve for acceleration:
a = (v^2 - u^2) / (2s)
Substituting the given values:
a = (0^2 - 15.6^2) / (2 * 0.0565)
a ≈ -45,870.88 m/s^2
Note: The negative sign indicates that the acceleration is in the opposite direction of the initial motion, as the automobile needs to decelerate to come to a stop.
Finally, we can calculate the net force using Newton's second law:
F = m * a
Substituting the given mass of the automobile (850 kg) and the calculated acceleration:
F = 850 kg * -45,870.88 m/s^2
F ≈ -39,000,250 N
So, approximately 39,000,250 Newtons of net force would be necessary to stop the automobile.