Understanding Uniform Acceleration: Calculating Initial Velocity and Acceleration
When dealing with the motion of objects, one common scenario is that of uniform acceleration, where the velocity changes at a constant rate. In this article, we will explore how to calculate the initial velocity and acceleration of a body given its average velocity and the distance it covers in a specific period.
Physical Laws and Problem Statement
According to the laws of motion, the average velocity of a body moving with uniform acceleration can be described using the formula:
Equation (1): (v_{avg} frac{v_i v_f}{2})
Where (v_{avg}) is the average velocity, (v_i) is the initial velocity, and (v_f) is the final velocity.
Another key equation in our toolkit is the distance formula for uniformly accelerated motion:
Equation (2): (S frac{1}{2} (v_i v_f) t)
Where (S) is the distance traveled, (v_i) is the initial velocity, (v_f) is the final velocity, and (t) is the time.
The object in our scenario covers an average velocity of (45 text{ cm/s}) for the first 10 seconds, and in the next 4 seconds, it covers (320 text{ cm}).
Step-by-Step Solution
First, we convert the given average velocity and distances into standard units for easier calculation:
Average velocity over 10 seconds: (45 text{ cm/s} 0.45 text{ m/s}) Distance covered in the next 4 seconds: (320 text{ cm} 3.20 text{ m})Next, we determine the average velocity and the total time for the 4 seconds of motion:
Step 1: Verify the average velocity over the 4 seconds.
The average velocity over these 4 seconds is described as the same as the final velocity of the motion in the first 10 seconds, which needs to be calculated.
Step 2: Calculate the final velocity at the end of the 10 seconds.
Since the average velocity is (0.45 text{ m/s}) and there was no specified change in velocity, we assume the final velocity at the end of the 10 seconds is the same as the average velocity, but we need to find it using the distance covered initially:
Equation (3): (v_f 0.45 text{ m/s})
Now, using the distance formula for the next 4 seconds, we can find the acceleration:
Step 3: Calculate the acceleration using the second equation.
Equation (4): (S frac{1}{2} (v_i v_f) t)
Given:
(S 3.20 text{ m}) (t 4 text{ s}) (v_f 0.45 text{ m/s}) (v_i ?)Substitute the known values into the equation to find (v_i):
Equation (5): (3.20 frac{1}{2} (v_i 0.45) times 4)
Solve for (v_i):
Equation (6): (3.20 2(v_i 0.45))
(1.60 v_i 0.45)
(v_i 1.15 text{ m/s})
Now that we know the initial velocity, (v_i 1.15 text{ m/s}), we can use the velocity formula to find the acceleration:
Equation (7): (v_f v_i at)
Where (v_f 0.45 text{ m/s}), (v_i 1.15 text{ m/s}), and (t 10 text{ s}).
Equation (8): (0.45 1.15 10a)
Solve for (a):
(0.45 - 1.15 10a)
(-0.70 10a)
(a -0.07 text{ m/s}^2)
Conclusion
By analyzing the motion of the body using the equations of motion and the provided data, we determined that the initial velocity (v_i) is (1.15 text{ m/s}) and the acceleration (a) is (-0.07 text{ m/s}^2).