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Locomotive of the mass M start moving so that its velocity varies according to the lobby equal to k under root s key is constant and as is the distance covered find the total work formed by all policies which are acting on locomotive during the first second after the beginning of motion?
Verified Answer
Locomotive of the mass M start moving so that its velocity varies acco...
Ds/dt = ksqrt(s)
s^(-1/2) ds = k dt
s^(1/2) / (1/2) = kt
2sqrt(s) = kt
s = k^2 t^2 / 4
a = d2s / dt2 = k^2 / 2
dW = F dS = ma dS
Since the force is constant,
W = ma integral(dS) = ma S = (mk^2 / 2) x (k^2 t^2 / 4)
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Locomotive of the mass M start moving so that its velocity varies acco...
Total Work Done on the Locomotive
To analyze the work done on the locomotive, we start with the given information that the velocity \( v \) varies as \( v = k \sqrt{s} \), where \( s \) is the distance covered and \( k \) is a constant.
1. Velocity and Acceleration Calculation
- The velocity equation can be differentiated to find acceleration \( a \):
\[
v = k \sqrt{s} \quad \Rightarrow \quad \frac{dv}{ds} = \frac{k}{2\sqrt{s}} \quad \Rightarrow \quad a = v \frac{dv}{ds} = k \sqrt{s} \cdot \frac{k}{2\sqrt{s}} = \frac{k^2}{2}
\]
- Therefore, the acceleration \( a \) is constant and equals \( \frac{k^2}{2} \).
2. Distance Covered in the First Second
- Using the kinematic equation for uniformly accelerated motion:
\[
s = ut + \frac{1}{2} a t^2
\]
- Since the initial velocity \( u = 0 \) at \( t = 0 \):
\[
s = \frac{1}{2} \cdot \frac{k^2}{2} \cdot (1^2) = \frac{k^2}{4}
\]
3. Work Done Calculation
- The work done \( W \) on the locomotive can be calculated using the formula:
\[
W = F \cdot s
\]
- The net force \( F \) acting on the locomotive is given by Newton's second law \( F = M \cdot a \):
\[
F = M \cdot \frac{k^2}{2}
\]
- Plugging in the values of \( F \) and \( s \):
\[
W = \left( M \cdot \frac{k^2}{2} \right) \cdot \left( \frac{k^2}{4} \right) = \frac{M k^4}{8}
\]
Conclusion
The total work done by all forces acting on the locomotive during the first second after the beginning of motion is \( \frac{M k^4}{8} \). This calculation incorporates the effects of both the constant acceleration and the distance traveled in that time frame.
To analyze the work done on the locomotive, we start with the given information that the velocity \( v \) varies as \( v = k \sqrt{s} \), where \( s \) is the distance covered and \( k \) is a constant.
1. Velocity and Acceleration Calculation
- The velocity equation can be differentiated to find acceleration \( a \):
\[
v = k \sqrt{s} \quad \Rightarrow \quad \frac{dv}{ds} = \frac{k}{2\sqrt{s}} \quad \Rightarrow \quad a = v \frac{dv}{ds} = k \sqrt{s} \cdot \frac{k}{2\sqrt{s}} = \frac{k^2}{2}
\]
- Therefore, the acceleration \( a \) is constant and equals \( \frac{k^2}{2} \).
2. Distance Covered in the First Second
- Using the kinematic equation for uniformly accelerated motion:
\[
s = ut + \frac{1}{2} a t^2
\]
- Since the initial velocity \( u = 0 \) at \( t = 0 \):
\[
s = \frac{1}{2} \cdot \frac{k^2}{2} \cdot (1^2) = \frac{k^2}{4}
\]
3. Work Done Calculation
- The work done \( W \) on the locomotive can be calculated using the formula:
\[
W = F \cdot s
\]
- The net force \( F \) acting on the locomotive is given by Newton's second law \( F = M \cdot a \):
\[
F = M \cdot \frac{k^2}{2}
\]
- Plugging in the values of \( F \) and \( s \):
\[
W = \left( M \cdot \frac{k^2}{2} \right) \cdot \left( \frac{k^2}{4} \right) = \frac{M k^4}{8}
\]
Conclusion
The total work done by all forces acting on the locomotive during the first second after the beginning of motion is \( \frac{M k^4}{8} \). This calculation incorporates the effects of both the constant acceleration and the distance traveled in that time frame.
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Locomotive of the mass M start moving so that its velocity varies according to the lobby equal to k under root s key is constant and as is the distance covered find the total work formed by all policies which are acting on locomotive during the first second after the beginning of motion?
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Locomotive of the mass M start moving so that its velocity varies according to the lobby equal to k under root s key is constant and as is the distance covered find the total work formed by all policies which are acting on locomotive during the first second after the beginning of motion? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Locomotive of the mass M start moving so that its velocity varies according to the lobby equal to k under root s key is constant and as is the distance covered find the total work formed by all policies which are acting on locomotive during the first second after the beginning of motion? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Locomotive of the mass M start moving so that its velocity varies according to the lobby equal to k under root s key is constant and as is the distance covered find the total work formed by all policies which are acting on locomotive during the first second after the beginning of motion?.
Locomotive of the mass M start moving so that its velocity varies according to the lobby equal to k under root s key is constant and as is the distance covered find the total work formed by all policies which are acting on locomotive during the first second after the beginning of motion? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Locomotive of the mass M start moving so that its velocity varies according to the lobby equal to k under root s key is constant and as is the distance covered find the total work formed by all policies which are acting on locomotive during the first second after the beginning of motion? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Locomotive of the mass M start moving so that its velocity varies according to the lobby equal to k under root s key is constant and as is the distance covered find the total work formed by all policies which are acting on locomotive during the first second after the beginning of motion?.
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