A stone is projected vertically upwards with velocity 40 m/s. Show that the stone will reach the point of projection with the same velocity.

hmax = (V^2-Vo^2)/2g.

hmax = (0-1600)/-19.6 = 81.6 m.

V^2 = Vo^2 + 2g*hmax.
V^2 = 0 + 19.6*81.6 = 1600
V = 40 m/s.

To show that the stone will reach the point of projection with the same velocity, we need to consider the principles of projectile motion and the laws of motion.

1. The first principle of projectile motion states that the horizontal motion of the object is independent of its vertical motion. This means that the only force acting horizontally on the stone is its initial velocity.

2. The second principle of motion, as stated by Newton's first law of motion, states that an object in motion will stay in motion with the same velocity unless acted upon by an external force.

Based on these principles, let's analyze the motion of the stone:

1. When the stone is projected vertically upwards, it will experience a downward force due to gravity. This force will gradually decrease its upward velocity.

2. As the stone reaches its highest point, its velocity will become zero momentarily. This is because its upward velocity is gradually reduced to zero due to the force of gravity acting in the opposite direction.

3. After reaching the highest point, the stone will start falling back towards the ground. Now, the force of gravity will act in the same direction as the stone's motion, causing it to accelerate downward.

4. Due to the symmetry of motion, the stone will cover the same distance on its way down as it did on its way up, assuming no air resistance or other external forces.

5. Since the stone covers the same distance on its way down, it will take the same amount of time to reach the point of projection as it did to reach its highest point.

6. As the stone reaches the point of projection, its velocity will be the same as it was when it was projected upwards, but in the opposite direction. This is because the downward force of gravity has increased its velocity to the same magnitude as the initial upward velocity.

Therefore, we can conclude that the stone will indeed reach the point of projection with the same velocity.

To show that the stone will reach the point of projection with the same velocity, we can use the laws of motion.

First, let's analyze the motion of the stone. We are given that the stone is projected vertically upwards with an initial velocity of 40 m/s. Since the stone is projected upwards, we can assume that the only force acting on it is gravity.

As the stone moves upward, gravity acts as a force to slow down its velocity. Eventually, the stone will reach its highest point and start to fall back downwards due to gravity.

At the highest point of its trajectory, the stone's velocity will become zero since it momentarily comes to a stop before reversing its direction. This is known as the peak or apex of the stone's motion.

Now, as the stone continues to fall back downwards, gravity continues to act on it, but in the opposite direction. This causes the stone to accelerate, increasing its velocity in the downward direction.

When the stone reaches the point of projection (the starting point), it will have the same velocity it had when it was initially projected upwards.

This can be explained by the symmetrical nature of projectile motion. Since the motion of the stone is symmetrical about the highest point of its trajectory, the velocity at any given height above the starting point on the way up will be the same as the velocity at the same height on the way down.

Therefore, we can conclude that the stone will reach the point of projection with the same velocity of 40 m/s.