A particle is traveling in a straight line at a constant speed of 29.9 m/s. Suddenly, a constant force of 14.5 N acts on it, bringing it to a stop in a distance of 58.2 m.
(a) What is the direction of the force?
(b) Determine the time it takes for the particle to come to a stop.
(c) What is its mass?
a = v^2/2s = 29.9^2/116.4 = 7.78
m = F/a = 14.5/7.78 = 1.88
to find t,
s = 1/2 at^2
clearly the force must act directly opposite its direction of travel.
To answer these questions, we can apply Newton's second law of motion which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
Let's break down the problem step by step:
(a) What is the direction of the force?
To determine the direction of the force, we need to consider the motion of the particle. The particle is traveling in a straight line and is brought to a stop, which means the force must act in the opposite direction of the particle's motion. Therefore, the force is directed opposite to the initial velocity vector of the particle.
(b) Determine the time it takes for the particle to come to a stop.
To find the time it takes for the particle to come to a stop, we can use the equation of motion:
v = u + at
Where:
v = final velocity (0 m/s since the particle comes to a stop)
u = initial velocity (29.9 m/s)
a = acceleration (which we'll solve for)
t = time
Rearranging the equation to solve for acceleration (a):
a = (v - u) / t
Since the particle comes to a stop, the final velocity (v) is 0. Substituting the given values:
0 = 29.9 m/s - (14.5 N / mass) * t
Simplifying the equation, we have:
29.9 m/s = (14.5 N / mass) * t
Now, we have one equation with two unknowns: mass and time.
(c) What is its mass?
To determine the mass of the particle, we need to find the value of (14.5 N / mass) from the equation we derived in question (b). To do this, we will first find the value of time (t).
Substituting the given values for the distance (58.2 m) and the initial velocity (29.9 m/s) into another equation of motion:
s = ut + 0.5at^2
Where:
s = distance (58.2 m)
u = initial velocity (29.9 m/s)
a = acceleration (which we'll solve for)
t = time
Rearranging the equation, we have:
0.5at^2 + ut - s = 0
Since the particle comes to a stop, the final velocity (v) is 0. Substituting the given values:
0.5a * t^2 + 29.9m/s * t - 58.2m = 0
This equation is a quadratic equation in terms of t. You can solve this equation using the quadratic formula:
t = (-b +/- sqrt(b^2 - 4ac)) / 2a
Where a = 0.5, b = 29.9 m/s, and c = -58.2 m.
Once you find the value of time (t), substitute it back into the equation we derived in question (b) to find the value of (14.5 N / mass). Rearranging that equation, we can solve for the mass (m).
By following these steps, you can determine the direction of the force, the time it takes for the particle to come to a stop, and the mass of the particle.