A(n) 2165.7 kg car is coasting along a level
road at 35.5 m/s. A constant braking force
is applied, such that the car is stopped in a
distance of 52.7 m.
What is the magnitude of the braking force?
Answer in units of N.
If the car stops in t seconds, the deceleration a is 35.5/t
The distance traveled is
s(t) = 35.5t - (35.5/2t)t^2 = 52.7
so, t = 2.97 seconds.
a = 35.5/2.97 = 11.953 m/s^2
F = ma = 2165.7 * 11.953 = 25,886 N
To find the magnitude of the braking force, we can use Newton's second law of motion, which states that the force (F) equals the mass (m) multiplied by the acceleration (a):
F = m * a
In this case, the car is being decelerated (braked) to a stop, so the acceleration will be a negative value.
First, let's calculate the acceleration:
Using the kinematic equation, we have:
v^2 = u^2 + 2as
where:
v = final velocity = 0 m/s (since the car comes to a stop)
u = initial velocity = 35.5 m/s
a = acceleration
s = distance traveled = 52.7 m
Rearranging the equation to solve for acceleration (a):
a = (v^2 - u^2) / (2s)
= (0^2 - 35.5^2) / (2 * 52.7)
a ≈ -21.67 m/s^2
Now we can substitute the values for mass (m = 2165.7 kg) and acceleration (a ≈ -21.67 m/s^2) into the force formula:
F = m * a
= 2165.7 kg * (-21.67 m/s^2)
Calculating the magnitude of the braking force:
F ≈ 46920.019 N
Therefore, the magnitude of the braking force is approximately 46920.019 N.
To find the magnitude of the braking force, we can use the equation of motion that relates acceleration, initial velocity, final velocity, and displacement:
v^2 = u^2 + 2as
where:
v = final velocity (0 m/s, as the car is stopped)
u = initial velocity (35.5 m/s)
s = displacement (52.7 m)
a = acceleration (unknown)
Rearranging the equation, we have:
a = (v^2 - u^2) / (2s)
Substituting the given values, we have:
a = (0^2 - 35.5^2) / (2 * 52.7)
a = (-1260.25) / (105.4)
a ≈ -11.95 m/s^2
The negative sign indicates that the car is experiencing retardation, or deceleration.
Now, to calculate the magnitude of the braking force, we can use Newton's second law of motion, which states:
force = mass * acceleration
The mass of the car is given as 2165.7 kg. Substituting the values, we have:
force = 2165.7 kg * (-11.95 m/s^2)
force ≈ -25891.8 N
Since force is a vector quantity, its magnitude is always positive. Therefore, we can take the absolute value of -25891.8 N to find the magnitude of the braking force.
Magnitude of the braking force ≈ 25891.8 N