A body with initial velocity 8.0 m/s moves along a straight line with constant acceleration and travels
640 m in 40 s. For the 40 s interval, find (a) the average velocity, (b) the final velocity, and (c) the
acceleration.
The Correct Answer and Explanation is :
Solution:
We are given the following information:
- Initial velocity, ( u = 8.0 \, \text{m/s} )
- Distance traveled, ( s = 640 \, \text{m} )
- Time interval, ( t = 40 \, \text{s} )
Using this information, let’s solve the problem step by step.
(a) Average velocity:
The formula for average velocity when the motion is uniformly accelerated is:
[
v_{\text{avg}} = \frac{s}{t}
]
Substituting the given values:
[
v_{\text{avg}} = \frac{640 \, \text{m}}{40 \, \text{s}} = 16 \, \text{m/s}
]
So, the average velocity is ( 16 \, \text{m/s} ).
(b) Final velocity:
For uniformly accelerated motion, we can use the equation:
[
v = u + at
]
where:
- ( v ) is the final velocity,
- ( a ) is the acceleration,
- ( t ) is the time.
We don’t know the acceleration yet, so first, we will use another kinematic equation to solve for ( a ):
[
s = ut + \frac{1}{2} a t^2
]
Substitute the known values:
[
640 = (8)(40) + \frac{1}{2} a (40)^2
]
Simplifying:
[
640 = 320 + 800a
]
Solving for ( a ):
[
320 = 800a \quad \Rightarrow \quad a = \frac{320}{800} = 0.4 \, \text{m/s}^2
]
Now, substitute ( a = 0.4 \, \text{m/s}^2 ) into the first equation for final velocity:
[
v = 8 + (0.4)(40) = 8 + 16 = 24 \, \text{m/s}
]
So, the final velocity is ( 24 \, \text{m/s} ).
(c) Acceleration:
From part (b), we already found the acceleration:
[
a = 0.4 \, \text{m/s}^2
]
Explanation (300 words):
In kinematics, average velocity is a measure of the total displacement covered in a given time interval. For uniform acceleration, the displacement (( s )) is related to time (( t )) and initial velocity (( u )) through equations of motion. Here, we use ( v_{\text{avg}} = \frac{s}{t} ) to find that the object has an average velocity of 16 m/s.
Next, to calculate the final velocity (( v )), we use the second equation of motion, ( s = ut + \frac{1}{2} at^2 ), where acceleration (( a )) is a key unknown. By substituting known values into this equation, we isolate ( a ) and determine it to be 0.4 m/s². This acceleration describes how much the velocity increases per second.
Finally, using the equation ( v = u + at ), we find the final velocity to be 24 m/s. The relationship between initial and final velocity is directly influenced by the acceleration, and in this case, the object gained 16 m/s over 40 seconds due to the constant acceleration.
These calculations showcase the core principles of uniformly accelerated motion, where acceleration remains constant, and both velocity and displacement follow predictable patterns over time.