kinematics?

Question

This is with respect to projectiles launched horizontally from a cliff :
when a projectile is launched with an initial velocity, is it true that it is equal to to the velocity just before it hits the ground?

under no air resistance conditions

Answer 1

It it not true for the scenario you describe above.

The projectile is launched horizontally from the cliff. As horizontal motion is unaffected by gravity, the horizontal component of the velocity will remain constant. (I think this is where you are getting confused).

When the projectile starts to move, it has zero vertical velocity. As it continues, it will begin to accelerate vertically under the influence of gravity. The vertical velocity will continue to increase as it falls.

The overall speed as it reaches the ground will be greater.

We calculate the overall speed by adding the horizontal and vertical components according to Pythagorus' theorem.

Answer 2

No it is not true. What is true is that the kinetic energy gained from the vertical drop just before impact will equal the potential energy of the projectile just before it is fired. In which case, mgh = 1/2 mv^2 and v = sqrt(2gh); where h is the height of the projectile above ground before it is fired. v is the vertical velocity of the projectile just before it hits the ground.

Note that ke = 1/2 mu^2; where u is the horizontal velocity of the projectile as it is fired from the muzzle and u = constant because there are no horizontal forces (like air drag) indicated in this problem. So we can conclude the horizontal velocity (u) is the same just before impact as it is upon firing from the cliff.

Thus, upon impact V^2 = v^2 + u^2; where V is the final velocity upon impact and it includes both u and v. Clearly the velocity just before impact V > u because v, the vertical velocity added due to the fall of the projectile, is also included in V.

Thus impact kinetic energy (KE) is greater than the kinetic energy (ke) upon firing. That is, KE/ke = 1/2 mV^2/1/2 mu^2 and KE = ke(V/u)^2 and KE > ke since V/u > 1.00

Answer 3

with no air resistance, yes this is true.

think of the forces acting on the projectile.

gravity is driving the projectile down towards the ground, but what is slowing the projectile down horizontally? only air, so if there is no air, then the projectile will have the same horizontal speed when it hits the ground as it had initially.