A particle is traveling in a straight line at a constant speed of 28.5 m/s. Suddenly, a constant force of 12.0 N acts on it, bringing it to a stop in a distance of 61.9 m.
(a)Determine the time it takes for the particle to come to a stop.
s
(b) What is its mass?
kg
To find the time it takes for the particle to come to a stop, we can use the equation:
v^2 = u^2 + 2as
Where:
v = final velocity (0 m/s, since the particle comes to a stop)
u = initial velocity (28.5 m/s)
a = acceleration
s = distance (61.9 m)
We need to find the acceleration, and then we can solve for time.
To find the acceleration, we can use Newton's second law of motion:
F = ma
Where:
F = force (12.0 N)
m = mass
a = acceleration
From the given values, we have a constant force of 12.0 N acting on the particle. We can equate this force to the mass (to be determined) multiplied by the acceleration.
12.0 N = m * a
Now, we can solve for acceleration (a):
a = 12.0 N / m
Substituting this value of acceleration into the first equation:
0^2 = (28.5 m/s)^2 + 2 * (12.0 N / m) * 61.9 m
Simplifying the equation:
0 = 812.25 m^2/s^2 + 24 * (12.0 N) * 61.9 m / m
To bring the equation to the form ax^2 + bx + c = 0, we can divide the entire equation by 812.25:
0 = 1 + 2 * 24 * 12.0 N * 61.9 m / (812.25 m^2/s^2)
Now, we have a quadratic equation of the form:
ax^2 + bx + c = 0
Where:
a = 1
b = 2 * 24 * 12.0 N * 61.9 m
c = 0
By factoring the quadratic equation, we can solve for x (time) and find the positive root, as the particle is coming to a stop.
Now, to find the mass of the particle, we can rearrange the equation F = ma:
12.0 N = m * a
We already know the acceleration (from solving part a), so we can substitute this value into the equation:
12.0 N = m * (acceleration)
Solving for mass (m):
m = 12.0 N / (acceleration)
Using the calculated value of acceleration, we can find the mass of the particle.