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A particle is moving in a straight line such that its velocity varies as v =v0 e-lemda t, where lemda and t are constant. Find the average velocity during the time interval in which the velocity decrease from v0 to v0/2?
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A particle is moving in a straight line such that its velocity varies ...
Introduction:
In this problem, we are given the velocity of a particle as a function of time. We need to find the average velocity of the particle during the time interval in which the velocity decreases from v0 to v0/2.
Derivation:
The velocity of the particle is given by v = v0 e^-λt
Let's find the time taken for the velocity to decrease from v0 to v0/2.
v = v0/2
v0 e^-λt = v0/2
e^-λt = 1/2
-λt = ln(1/2)
-λt = -ln2
t = ln2/λ
Now, we need to find the average velocity of the particle during the time interval from t=0 to t=ln2/λ.
Solution:
Step 1: Find the displacement of the particle during the time interval from t=0 to t=ln2/λ.
The displacement of the particle during this time interval is given by:
Δx = ∫v0/2v0 v(t) dt
= ∫v0/2v0 v0 e^-λt dt
= -v0/λ [e^-λt]v0/2v0
= -v0/λ [e^-(λln2/λ) - 1]
= -v0/λ [1/2 - 1]
= v0/λ [1/2]
Δx = v0/2λ
Step 2: Find the time taken for the particle to travel from v0 to v0/2.
The time taken for the particle to travel from v0 to v0/2 is given by:
t = ln2/λ
Step 3: Find the average velocity of the particle during the time interval from t=0 to t=ln2/λ.
The average velocity of the particle during this time interval is given by:
v_avg = Δx / t
v_avg = (v0/2λ) / (ln2/λ)
v_avg = v0/2 ln2
Conclusion:
Therefore, the average velocity of the particle during the time interval in which the velocity decreases from v0 to v0/2 is v0/2 ln2.
In this problem, we are given the velocity of a particle as a function of time. We need to find the average velocity of the particle during the time interval in which the velocity decreases from v0 to v0/2.
Derivation:
The velocity of the particle is given by v = v0 e^-λt
Let's find the time taken for the velocity to decrease from v0 to v0/2.
v = v0/2
v0 e^-λt = v0/2
e^-λt = 1/2
-λt = ln(1/2)
-λt = -ln2
t = ln2/λ
Now, we need to find the average velocity of the particle during the time interval from t=0 to t=ln2/λ.
Solution:
Step 1: Find the displacement of the particle during the time interval from t=0 to t=ln2/λ.
The displacement of the particle during this time interval is given by:
Δx = ∫v0/2v0 v(t) dt
= ∫v0/2v0 v0 e^-λt dt
= -v0/λ [e^-λt]v0/2v0
= -v0/λ [e^-(λln2/λ) - 1]
= -v0/λ [1/2 - 1]
= v0/λ [1/2]
Δx = v0/2λ
Step 2: Find the time taken for the particle to travel from v0 to v0/2.
The time taken for the particle to travel from v0 to v0/2 is given by:
t = ln2/λ
Step 3: Find the average velocity of the particle during the time interval from t=0 to t=ln2/λ.
The average velocity of the particle during this time interval is given by:
v_avg = Δx / t
v_avg = (v0/2λ) / (ln2/λ)
v_avg = v0/2 ln2
Conclusion:
Therefore, the average velocity of the particle during the time interval in which the velocity decreases from v0 to v0/2 is v0/2 ln2.
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A particle is moving in a straight line such that its velocity varies ...
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A particle is moving in a straight line such that its velocity varies as v =v0 e-lemda t, where lemda and t are constant. Find the average velocity during the time interval in which the velocity decrease from v0 to v0/2?
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A particle is moving in a straight line such that its velocity varies as v =v0 e-lemda t, where lemda and t are constant. Find the average velocity during the time interval in which the velocity decrease from v0 to v0/2? for NEET 2024 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about A particle is moving in a straight line such that its velocity varies as v =v0 e-lemda t, where lemda and t are constant. Find the average velocity during the time interval in which the velocity decrease from v0 to v0/2? covers all topics & solutions for NEET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A particle is moving in a straight line such that its velocity varies as v =v0 e-lemda t, where lemda and t are constant. Find the average velocity during the time interval in which the velocity decrease from v0 to v0/2?.
A particle is moving in a straight line such that its velocity varies as v =v0 e-lemda t, where lemda and t are constant. Find the average velocity during the time interval in which the velocity decrease from v0 to v0/2? for NEET 2024 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about A particle is moving in a straight line such that its velocity varies as v =v0 e-lemda t, where lemda and t are constant. Find the average velocity during the time interval in which the velocity decrease from v0 to v0/2? covers all topics & solutions for NEET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A particle is moving in a straight line such that its velocity varies as v =v0 e-lemda t, where lemda and t are constant. Find the average velocity during the time interval in which the velocity decrease from v0 to v0/2?.
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