Starting from rest a particle confined to move along a straight line is accelerated at a rate of 5m/s. Describe the motion of the particle.
The velocity of the particle increases by a constant 5 m/s each second in the same direction. If we graph velocity v. time, we get a straight line with a positive slope. This line goes through the origin.
To describe the motion of a particle accelerated at a constant rate, we can use the equations of motion. In this case, the particle begins from rest, meaning its initial velocity is zero.
Given:
Initial velocity (u) = 0 m/s
Acceleration (a) = 5 m/s²
We can use the following equations of motion:
1. Final velocity (v) = u + at
2. Distance traveled (s) = ut + (1/2)at²
Let's solve for the final velocity (v) first:
Substituting the given values into equation (1):
v = 0 + 5t
v = 5t m/s
Now, let's find the distance traveled (s):
Substituting the given values into equation (2):
s = 0t + (1/2)(5t²)
s = (1/2) * 5t²
s = (5/2)t² m
From these equations, we can see that the particle's velocity (v) will be directly proportional to time (t) and the distance traveled (s) will be directly proportional to time squared (t²).
Therefore, as time progresses, the particle's velocity will continuously increase at a constant rate of 5 m/s, while the distance traveled will become increasingly larger as time is squared.
In summary, the motion of the particle will be one of continuously increasing velocity and an increasing distance as a function of time squared.
To describe the motion of the particle, we can use kinematic equations. Given that the particle starts from rest and has a constant acceleration of 5 m/s², we can determine its motion using the following steps:
Step 1: Determine the initial velocity (u) of the particle.
Given that the particle starts from rest, the initial velocity (u) is 0 m/s.
Step 2: Determine the final velocity (v) of the particle.
To find the final velocity (v), we can use the equation:
v = u + at
where:
v = final velocity
u = initial velocity
a = acceleration
t = time
Since the initial velocity (u) is 0 m/s and the acceleration (a) is 5 m/s², we can substitute the values into the equation:
v = 0 + (5 m/s²) * t
Simplifying the equation, we get:
v = 5t
Step 3: Determine the position (s) of the particle.
To find the position (s) of the particle, we can use the equation:
s = ut + (1/2)at²
where:
s = displacement or position
u = initial velocity
t = time
a = acceleration
Since the initial velocity (u) is 0 m/s and the acceleration (a) is 5 m/s², we can substitute the values into the equation:
s = (1/2) (5 m/s²) * t²
Simplifying the equation, we get:
s = (1/2) * 5t²
s = (5/2) * t²
or
s = (2.5) * t²
So, the motion of the particle can be described by the equations:
v = 5t
s = (2.5) * t²
These equations give the relationship between time and the particle's velocity (v) and position (s).