A projectile is projected with velocity u= ai+bj from ground if acceleration due ti gravity is g, then the change in velocity of the projectile in second of projection is

To analyze the change in velocity of a projectile projected with an initial velocity of \( \mathbf{u} = a\mathbf{i} + b\mathbf{j} \) from the ground, we need to consider the effects of gravity on the projectile's motion. The acceleration due to gravity acts downward, affecting only the vertical component of the velocity. Let's break this down step by step.

Understanding Initial Velocity Components

The initial velocity \( \mathbf{u} \) can be separated into its horizontal and vertical components:

• Horizontal component: \( u_x = a \)
• Vertical component: \( u_y = b \)

Acceleration Due to Gravity

Gravity acts downward with an acceleration of \( g \). This means that while the horizontal component of the velocity remains constant (since there is no horizontal acceleration), the vertical component will change due to the influence of gravity.

Calculating Change in Velocity

At any time \( t \), the vertical component of the velocity can be expressed as:

Vertical velocity at time t: \( v_y = u_y - gt = b - gt \)

Here, we subtract \( gt \) because gravity acts in the opposite direction to the initial vertical velocity.

Change in Velocity After 1 Second

To find the change in velocity after 1 second of projection, we can evaluate the vertical velocity at \( t = 1 \) second:

Vertical velocity after 1 second: \( v_y(1) = b - g \cdot 1 = b - g \)

The horizontal component remains unchanged, so:

Horizontal velocity: \( v_x = a \)

Overall Change in Velocity

The change in velocity \( \Delta \mathbf{v} \) after 1 second can be expressed as:

Change in velocity: \( \Delta \mathbf{v} = \mathbf{v}(1) - \mathbf{u} \)

Where \( \mathbf{v}(1) = a\mathbf{i} + (b - g)\mathbf{j} \). Thus, we have:

Change in velocity: \( \Delta \mathbf{v} = (a\mathbf{i} + (b - g)\mathbf{j}) - (a\mathbf{i} + b\mathbf{j}) \)

This simplifies to:

Final expression: \( \Delta \mathbf{v} = 0\mathbf{i} - g\mathbf{j} = -g\mathbf{j} \)

Conclusion

Therefore, the change in velocity of the projectile after 1 second of projection is \( -g\mathbf{j} \), indicating that the vertical component of the velocity has decreased by \( g \) due to the effect of gravity. The horizontal component remains unchanged throughout the motion. This analysis highlights how gravity influences the vertical motion of projectiles while leaving horizontal motion unaffected.