Projectile Motion

Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration due to gravity. The path followed by a projectile is called its trajectory.

Deriving the Expression for Maximum Height Attained

To derive the expression for the maximum height attained by a projectile, we need to consider the motion of the projectile in the vertical direction. The projectile is subject to the acceleration due to gravity (g) in the vertical direction. The initial velocity of the projectile can be resolved into horizontal and vertical components.

Let u be the initial velocity of the projectile, θ be the angle of projection with the horizontal, and h be the maximum height attained by the projectile.

Using the equations of motion, we have:

v = u + gt (where v is the final velocity)

At maximum height, v = 0, and t is the time taken to reach the maximum height.

So, 0 = u sin θ - gt

Solving for t, we get:

t = u sin θ / g

Now, using the equation of motion, we have:

h = ut sin θ - 1/2 gt^2

Substituting the value of t, we get:

h = u^2 sin^2 θ / 2g

Therefore, the expression for the maximum height attained by a projectile is:

h = u^2 sin^2 θ / 2g

Deriving the Expression for Horizontal Range

To derive the expression for the horizontal range, we need to consider the motion of the projectile in the horizontal direction. The projectile is subject to no acceleration in the horizontal direction, and it moves with uniform velocity.

Let R be the horizontal range of the projectile.

Using the equations of motion, we have:

R = ut cos θ

Substituting the value of t from above, we get:

R = u^2 sin 2θ / g

Therefore, the expression for the horizontal range of a projectile is:

R = u^2 sin 2θ / g

Deriving the Expression for Time of Flight

The time of flight is the time taken by the projectile to reach the ground. To derive the expression for the time of flight, we need to consider the motion of the projectile in the vertical direction.

Using the equations of motion, we have:

y = ut sin θ - 1/2 gt^2

where y is the displacement of the projectile in the vertical direction.

At the highest point, y = h, and t is the time taken to reach the highest point.

So, h = u^2 sin^2 θ / 2g

Solving for t, we get:

t = 2u sin θ / g

Therefore, the expression for the time of flight of a projectile is:

t = 2u sin θ / g

Conclusion

In conclusion, the expressions for the maximum height attained, horizontal range, and time of flight of a projectile are derived by analyzing the motion of the projectile in the vertical and horizontal directions. These expressions are useful in analyzing the motion of projectiles in various applications, such as ballistics and sports.