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51. A ball of mass m is dropped from a tall building with zero initial velocity. In addition to gravity, the ball experiences a damping force of the form -yu, where v is its instantaneous velocity and y is a constant. Given the values m = 10 kg, y=10kg/s, and g = 10m/s², the distance trav- elled (in metres) in time t in seconds, is
(a) 10(1+1-e)
(c) 5t²-(1-e')
(b) 10(-1+e')
(d) 5t²?
(a) 10(1+1-e)
(c) 5t²-(1-e')
(b) 10(-1+e')
(d) 5t²?
Most Upvoted Answer
51. A ball of mass m is dropped from a tall building with zero initial...
Understanding the Problem
The ball is affected by two forces: gravitational force and damping force. The gravitational force acts downwards, while the damping force opposes the velocity of the ball.
Key Parameters
- Mass (m) = 10 kg
- Damping constant (y) = 10 kg/s
- Gravitational acceleration (g) = 10 m/s²
Equation of Motion
The equation of motion can be derived from Newton's second law. The net force acting on the ball when dropped is:
F = mg - yv
This leads to the second-order differential equation:
m * dv/dt = mg - yv
Solving the Differential Equation
Rearranging the above equation yields:
dv/dt + (y/m)v = g
Substituting the values:
dv/dt + (10/10)v = 10
This simplifies to:
dv/dt + v = 10
Finding the Velocity
Using integrating factors or separation of variables, we solve for velocity (v):
v(t) = 10(1 - e^(-t))
Calculating the Distance
To find the distance traveled (s), we integrate the velocity:
s(t) = ∫v(t) dt = ∫(10(1 - e^(-t))) dt
This results in:
s(t) = 10t + 10e^(-t) + C
Given the initial condition (s(0) = 0), we find the constant C = -10.
Thus, the distance becomes:
s(t) = 10t + 10e^(-t) - 10
This can be simplified to:
s(t) = 10(-1 + e^(-t))
Final Answer
The correct option is (b) 10(-1 + e^(-t)).
The ball is affected by two forces: gravitational force and damping force. The gravitational force acts downwards, while the damping force opposes the velocity of the ball.
Key Parameters
- Mass (m) = 10 kg
- Damping constant (y) = 10 kg/s
- Gravitational acceleration (g) = 10 m/s²
Equation of Motion
The equation of motion can be derived from Newton's second law. The net force acting on the ball when dropped is:
F = mg - yv
This leads to the second-order differential equation:
m * dv/dt = mg - yv
Solving the Differential Equation
Rearranging the above equation yields:
dv/dt + (y/m)v = g
Substituting the values:
dv/dt + (10/10)v = 10
This simplifies to:
dv/dt + v = 10
Finding the Velocity
Using integrating factors or separation of variables, we solve for velocity (v):
v(t) = 10(1 - e^(-t))
Calculating the Distance
To find the distance traveled (s), we integrate the velocity:
s(t) = ∫v(t) dt = ∫(10(1 - e^(-t))) dt
This results in:
s(t) = 10t + 10e^(-t) + C
Given the initial condition (s(0) = 0), we find the constant C = -10.
Thus, the distance becomes:
s(t) = 10t + 10e^(-t) - 10
This can be simplified to:
s(t) = 10(-1 + e^(-t))
Final Answer
The correct option is (b) 10(-1 + e^(-t)).
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51. A ball of mass m is dropped from a tall building with zero initial velocity. In addition to gravity, the ball experiences a damping force of the form -yu, where v is its instantaneous velocity and y is a constant. Given the values m = 10 kg, y=10kg/s, and g = 10m/s², the distance trav- elled (in metres) in time t in seconds, is(a) 10(1+1-e)(c) 5t²-(1-e')(b) 10(-1+e')(d) 5t²?
Question Description
51. A ball of mass m is dropped from a tall building with zero initial velocity. In addition to gravity, the ball experiences a damping force of the form -yu, where v is its instantaneous velocity and y is a constant. Given the values m = 10 kg, y=10kg/s, and g = 10m/s², the distance trav- elled (in metres) in time t in seconds, is(a) 10(1+1-e)(c) 5t²-(1-e')(b) 10(-1+e')(d) 5t²? for GATE Physics 2025 is part of GATE Physics preparation. The Question and answers have been prepared according to the GATE Physics exam syllabus. Information about 51. A ball of mass m is dropped from a tall building with zero initial velocity. In addition to gravity, the ball experiences a damping force of the form -yu, where v is its instantaneous velocity and y is a constant. Given the values m = 10 kg, y=10kg/s, and g = 10m/s², the distance trav- elled (in metres) in time t in seconds, is(a) 10(1+1-e)(c) 5t²-(1-e')(b) 10(-1+e')(d) 5t²? covers all topics & solutions for GATE Physics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 51. A ball of mass m is dropped from a tall building with zero initial velocity. In addition to gravity, the ball experiences a damping force of the form -yu, where v is its instantaneous velocity and y is a constant. Given the values m = 10 kg, y=10kg/s, and g = 10m/s², the distance trav- elled (in metres) in time t in seconds, is(a) 10(1+1-e)(c) 5t²-(1-e')(b) 10(-1+e')(d) 5t²?.
51. A ball of mass m is dropped from a tall building with zero initial velocity. In addition to gravity, the ball experiences a damping force of the form -yu, where v is its instantaneous velocity and y is a constant. Given the values m = 10 kg, y=10kg/s, and g = 10m/s², the distance trav- elled (in metres) in time t in seconds, is(a) 10(1+1-e)(c) 5t²-(1-e')(b) 10(-1+e')(d) 5t²? for GATE Physics 2025 is part of GATE Physics preparation. The Question and answers have been prepared according to the GATE Physics exam syllabus. Information about 51. A ball of mass m is dropped from a tall building with zero initial velocity. In addition to gravity, the ball experiences a damping force of the form -yu, where v is its instantaneous velocity and y is a constant. Given the values m = 10 kg, y=10kg/s, and g = 10m/s², the distance trav- elled (in metres) in time t in seconds, is(a) 10(1+1-e)(c) 5t²-(1-e')(b) 10(-1+e')(d) 5t²? covers all topics & solutions for GATE Physics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 51. A ball of mass m is dropped from a tall building with zero initial velocity. In addition to gravity, the ball experiences a damping force of the form -yu, where v is its instantaneous velocity and y is a constant. Given the values m = 10 kg, y=10kg/s, and g = 10m/s², the distance trav- elled (in metres) in time t in seconds, is(a) 10(1+1-e)(c) 5t²-(1-e')(b) 10(-1+e')(d) 5t²?.
Solutions for 51. A ball of mass m is dropped from a tall building with zero initial velocity. In addition to gravity, the ball experiences a damping force of the form -yu, where v is its instantaneous velocity and y is a constant. Given the values m = 10 kg, y=10kg/s, and g = 10m/s², the distance trav- elled (in metres) in time t in seconds, is(a) 10(1+1-e)(c) 5t²-(1-e')(b) 10(-1+e')(d) 5t²? in English & in Hindi are available as part of our courses for GATE Physics.
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