Suppose a ball of mass m is thrown vertically upward with an initial speed v, its
speed decreases continuously till it becomes zero. Thereafter, the ball begins to
fall downward and attains the speed v again before striking the ground. It
implies that the magnitude of initial and final momentums of the ball are same.
Yet, it is not an example of conservation of momentum. Explain why ?
In deep space (no gravity), the bolt (arrow)
of a crossbow accelerates at 204 m/s
2
and
attains a speed of 130 m/s when it leaves the
bow.
For how long is it accelerated?
Answer in units of s
To understand why this scenario is not an example of conservation of momentum, we need to analyze the situation in detail.
The momentum of an object is the product of its mass and its velocity. In this case, we have a ball of mass "m" that is thrown vertically upward with an initial speed "v". As the ball moves upward, its speed decreases continuously until it becomes zero at the highest point of its trajectory. At this point, the ball changes direction and starts falling downward.
When the ball reaches its highest point, its momentum is zero because the velocity is zero. As the ball falls downward, its speed increases until it reaches the same initial speed "v" with which it was thrown upward. Finally, the ball strikes the ground.
Now, let's examine the initial and final momenta of the ball:
1. Initial momentum: When the ball is thrown upward, its initial momentum is given by the equation p_initial = m * v, where "m" is the mass of the ball and "v" is its initial velocity. So, the magnitude of the initial momentum is m * v.
2. Final momentum: When the ball strikes the ground, its final momentum is given by the equation p_final = m * v, where "m" is the mass of the ball and "v" is the same initial velocity as before. So, the magnitude of the final momentum is m * v.
From the above analysis, we can see that the magnitudes of the initial and final momenta of the ball are indeed the same (m * v). However, this does not imply conservation of momentum.
Conservation of momentum states that the total momentum of a system remains constant if no external forces act on it. In this case, we have neglected any external forces such as air resistance. But, as the ball is thrown upward and then falls downward, gravity acts on it as an external force.
Gravity is a constant force that acts vertically downward. It continuously affects the ball's motion, changing its velocity and momentum. Although the initial and final momenta have the same magnitude, they are not equal in terms of direction. The initial momentum is upward, while the final momentum is downward.
Therefore, the scenario described does not satisfy the condition for conservation of momentum because external forces, like gravity, are acting on the system.