Concept explainers
(a)
Speed of the projectile when it leaves the barrel.
(a)
Answer to Problem 21P
Speed of the projectile when it leaves the barrel is
Explanation of Solution
In the system elastic potential energy is due to the spring and as friction force is acting on the ball total energy of the system is due to kinetic energy, eleastic potential energy and energy due to frictional force.
When the cannon is fired initial speed of the ball is zero as initially ball is at rest so, initial kinetic energy of the system is zero and spring finally returns to equilibriam so final elastic potential energy is also zero.
Write the expression for change in kinetic energy of the system.
Here,
Write the expression for change inelastic potential energy.
Here,
Write the expression for total internal energy.
Here,
Total energy of the system remains conserved therefore sum of all energies is equal to zero.
Write the expression for change conservation of energy for the system.
Substitute
Substitute
Multiply above equation by
Rearrange the above equation for
Conclusion:
Substitute
Thus, Speed of the projectile when it leaves the barrel is
(b)
The distance at which the ball has maximum speed.
(b)
Answer to Problem 21P
The distance at which the ball has maximum speed is
Explanation of Solution
The ball has maximum speed when there is no further acceleration. It happens when the force due to spring acting on the ball is equal to frictional force.
Here,
Write the expression for force due to spring on the ball.
Here,
Substitute
Rearrange the above equation for
Write the expression for distance where the speed is maximum.
Here,
Conclusion:
Substitute
Substitute
Thus, the distance at which the ball has maximum speed is
(c)
Maximum speed at the point asked in part (b).
(c)
Answer to Problem 21P
Maximum speed at the point asked in part (b) is
Explanation of Solution
At this point initial speed of the ball is zero as initially ball is at rest so, initial kinetic energy of the system is zero .
Substitute
Rearrange above equation.
Multiply above equation by
Rearrange above equation for
Conclusion:
Substitute
Thus, the maximum speed at the point asked in part (b) is
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