Understanding Projectile Motion
When a body is thrown upwards at an angle, its motion can be broken down into two components: horizontal and vertical. The initial velocity can be resolved into these components using the angle theta.
Velocity Components
- The horizontal component (Vx) is given by:
Vx = v * cos(theta)
This component remains constant throughout the motion since there is no horizontal acceleration (ignoring air resistance).
- The vertical component (Vy) is given by:
Vy = v * sin(theta)
This component changes due to gravitational acceleration acting downwards.
Calculating Vertical Velocity After Time t
The vertical velocity after time t can be calculated using the formula:
Vy(t) = Vy - g * t
Where g is the acceleration due to gravity (approximately 9.81 m/s²).
Substituting the initial vertical velocity:
Vy(t) = (v * sin(theta)) - (g * t)
Magnitude of Velocity After Time t
To find the magnitude of the velocity after time t, we consider both components:
- The horizontal velocity remains constant:
Vx = v * cos(theta)
- The vertical velocity at time t is:
Vy(t) = (v * sin(theta)) - (g * t)
Using these two components, the magnitude of the total velocity (V) can be determined using the Pythagorean theorem:
V = √(Vx² + Vy(t)²)
V = √((v * cos(theta))² + ((v * sin(theta)) - (g * t))²)
Conclusion
The magnitude of the velocity after t seconds is derived from the constant horizontal component and the changing vertical component, allowing for a comprehensive understanding of the body's motion in projectile motion scenarios.