To find the projectile's time of flight, maximum height, and range, we calculated its velocity components, used motion equations, and got a flight time of approximately 4.33 seconds, maximum height around 22.86 meters, and range close to 91.87 meters.
Step-by-step explanation:
Projectile Motion Calculations
To calculate the time of flight, maximum height, and horizontal range of a projectile launched at an initial velocity of 30 m/s at a 45-degree angle, we'll break down the initial velocity into horizontal and vertical components using trigonometric functions. Then we'll use the equations of projectile motion to determine each parameter.
Time of Flight: We use the equation δy = V0y × t + 0.5 × g × t² where δy is the vertical displacement (0 for a return to ground level), V0y is the initial vertical velocity, and g is the acceleration due to gravity (9.8 m/s²). For a 45-degree launch angle, the vertical component of the initial velocity is V0 × sin(45°) = 30 m/s × √(1/2) ≈ 21.21 m/s. The time of flight is then 2 × V0y / g ≈ 2 × (21.21 m/s) / (9.8 m/s²) ≈ 4.33 seconds.
Maximum Height: The maximum height is calculated using V²y = V0²y − 2 × g × h, where Vy is the final vertical velocity (0 at maximum height), and h is the maximum height. Plugging in V0y, we get 0 = (21.21 m/s)² − 2 × (9.8 m/s²) × h. Solving for h yields a maximum height of about 22.86 meters.
Horizontal Range: The horizontal range is calculated using δx = V0x × t, where δx is the horizontal distance, and V0x is the initial horizontal velocity, which for a 45-degree angle is the same as V0y ≈ 21.21 m/s. Using the time of flight previously calculated, the range is V0x × t ≈ 21.21 m/s × 4.33 s ≈ 91.87 meters.
The actions above help in finding the time of flight, maximum height, and horizontal range of a projectile motion.
The probable question can be: please show work if possible
A projectile is launched from the ground with an initial velocity of 30 m/s at an angle of 45 degrees above the horizontal. Neglecting air resistance, calculate the following:
1. The time of flight.
2. The maximum height reached by the projectile.
3. The horizontal range of the projectile.