To calculate the range and velocity of a projectile at a specific time, we can break down the motion into horizontal and vertical components. Given that the initial velocity of the projectile is u = 30i + 40j m/s, we can use the following formulas:
- Horizontal motion: The horizontal motion of the projectile is unaffected by gravity, so the velocity remains constant throughout the entire trajectory. The horizontal component of velocity (Vx) is given by: Vx = uₓ, where uₓ is the x-component of the initial velocity.
In this case, uₓ = 30 m/s.
- Vertical motion: The vertical motion is influenced by gravity. The initial vertical component of velocity (Vy) is given by the y-component of the initial velocity: Vy = uᵧ, where uᵧ is the y-component of the initial velocity.
In this case, uᵧ = 40 m/s.
The acceleration due to gravity is typically denoted by "g" and has a value of approximately 9.8 m/s² (assuming no air resistance).
Using these values, we can calculate the range and velocity at t = 4s:
- Range (horizontal distance traveled): The range is the horizontal distance covered by the projectile. It can be calculated using the formula: Range = Vx * time, where time is the given time of flight.
In this case, time = 4s and Vx = uₓ = 30 m/s. Range = Vx * time = 30 m/s * 4 s = 120 m.
Therefore, the range of the projectile at t = 4s is 120 meters.
- Velocity at t = 4s: To find the velocity at t = 4s, we need to consider both the horizontal and vertical components of velocity. The magnitude of the velocity (V) can be calculated using the Pythagorean theorem: V = √(Vx² + Vy²), where Vx and Vy are the horizontal and vertical components of velocity, respectively.
In this case, Vx = 30 m/s and Vy = uᵧ - g * time = 40 m/s - 9.8 m/s² * 4 s = 40 m/s - 39.2 m/s = 0.8 m/s.
V = √(30 m/s)² + (0.8 m/s)² = √(900 m²/s² + 0.64 m²/s²) = √900.64 m²/s² = 30.01 m/s (approximately).
Therefore, the velocity of the projectile at t = 4s is approximately 30.01 m/s.