a ball is projected with a velocity 20 ms at an angle to the horizontal in order to have the maximum range its velocity at the highest point must be

To determine the velocity of a ball at its highest point when projected at an angle for maximum range, we need to consider the physics of projectile motion. When a projectile reaches its peak, its vertical component of velocity becomes zero, while the horizontal component remains unchanged throughout the motion.

Understanding Projectile Motion

Projectile motion can be broken down into two components: horizontal and vertical. The initial velocity can be split into these two components based on the angle of projection.

• Horizontal Component (Vx): This is calculated using the formula: Vx = V * cos(θ)
• Vertical Component (Vy): This is calculated using the formula: Vy = V * sin(θ)

Here, V is the initial velocity (20 m/s), and θ is the angle of projection. For maximum range, the optimal angle is 45 degrees, which simplifies our calculations.

Calculating Components at 45 Degrees

When the angle is 45 degrees, the sine and cosine values are equal:

• cos(45°) = sin(45°) = √2/2 ≈ 0.707

Now, we can calculate the horizontal and vertical components:

• Vx = 20 m/s * 0.707 ≈ 14.14 m/s
• Vy = 20 m/s * 0.707 ≈ 14.14 m/s

Velocity at the Highest Point

At the highest point of its trajectory, the vertical component of the velocity (Vy) becomes zero because the ball momentarily stops rising before it starts to fall back down. However, the horizontal component (Vx) remains constant throughout the flight. Therefore, at the peak, the total velocity of the ball is solely due to the horizontal component.

Final Calculation

Since the vertical component is zero at the highest point, the velocity of the ball at this point is:

• V = Vx = 14.14 m/s

In summary, when a ball is projected with an initial velocity of 20 m/s at an angle of 45 degrees for maximum range, its velocity at the highest point is approximately 14.14 m/s, directed horizontally. This illustrates the fascinating nature of projectile motion, where the components of velocity play crucial roles in determining the trajectory and range of the projectile.