if h be the maximum height of a projectile moving under the gravitatio...
Proof of the Velocity of Projection of a Projectile
Maximum Height of a Projectile
Let's first understand the maximum height of a projectile moving under the gravitational field of earth:
- The maximum height of a projectile is achieved at the highest point of its trajectory.
- At this point, the vertical component of velocity becomes zero.
- Using the formula for displacement in the vertical direction, we can find the maximum height:
Maximum height, h = (u^2 * sin^2(theta))/2g
Velocity of Projection of a Projectile
Now, let's prove that the velocity of projection of a projectile is root2gh / sin theta:
- Using the formula for time of flight, we can express the time taken by a projectile to reach its maximum height:
Time of flight, t = 2u * sin(theta) / g
- Using the formula for displacement in the horizontal direction, we can find the range of the projectile:
Range, R = u^2 * sin(2*theta) / g
- Using the formula for average velocity, we can express the average velocity of the projectile as:
Average velocity, Vavg = R / t = u * sin(2*theta) / 2
- Now, using the formula for the vertical component of velocity at the highest point:
Vertical component of velocity at highest point, V = u * sin(theta) - gt
- Since the vertical component of velocity becomes zero at the highest point, we can equate V to zero and solve for u:
u = {gt}/{sin(theta)}
- Substituting this value of u in the formula for average velocity, we get:
Vavg = {g * sin(2*theta)}/{2 * sin(theta)}
- Using the formula for the maximum height of a projectile, we can express g in terms of h:
g = 2h / (sin(theta))^2
- Substituting this value of g in the formula for average velocity, we get:
Vavg = {2h * sin(2*theta)}/{(sin(theta))^3}
- Now, using the formula for the horizontal component of velocity, we can express u in terms of Vavg:
u = Vavg * 2 / sin(2*theta)
- Substituting this value of u in the formula for the maximum height of a projectile, we get:
h = (Vavg^2 * sin^2(theta))/2g = (Vavg^2 * sin^2(theta))/4h