What is the maximum height reached by a projectile fired at an angle of 53° above the horizontal with an initial velocity of 50 m/s? Also, what is the position and velocity of the projectile 6 seconds later?
Maximum Height Calculation:
The maximum height reached by the projectile can be calculated using the kinematic equation for vertical motion. The formula to find the maximum height (h_max) is:
h_max = (V₀² * sin²θ) / (2 * g)
where V₀ is the initial velocity, θ is the launch angle, and g is the acceleration due to gravity.
Substitute the given values:
V₀ = 50 m/s
θ = 53° (converted to radians: 53° * π/180 ≈ 0.9273 rad)
g = 9.8 m/s²
Calculating:
h_max = (50² * sin²(0.9273)) / (2 * 9.8) ≈ 67.35 m
Therefore, the maximum height reached by the projectile is approximately 67.35 meters.
Position and Velocity 6 Seconds Later:
To find the position and velocity of the projectile 6 seconds later, we analyze its horizontal and vertical motion separately.
For the horizontal motion:
x = V₀ * cosθ * t
Substitute the given values:
V₀ = 50 m/s
θ = 53° (converted to radians: 53° * π/180 ≈ 0.9273 rad)
t = 6 s
Calculating:
x = 50 * cos(0.9273) * 6 ≈ 155.33 m
For the vertical motion:
y = V₀ * sinθ * t - (1/2) * g * t²
Substitute the given values:
V₀ = 50 m/s
θ = 53° (converted to radians: 53° * π/180 ≈ 0.9273 rad)
g = 9.8 m/s²
t = 6 s
Calculating:
y = 50 * sin(0.9273) * 6 - (1/2) * 9.8 * 6² ≈ 41.47 m
Therefore, 6 seconds later, the projectile will be at a horizontal position of approximately 155.33 meters and a vertical position of approximately 41.47 meters. The velocity at this time is approximately 19.98 m/s horizontally and -40.04 m/s vertically.
Projectile Motion: Projectile motion refers to the motion of an object that is projected into the air and is subject to the force of gravity. The path followed by a projectile is called its trajectory.
Maximum Height Calculation: By using the kinematic equation for vertical motion, we can determine the maximum height reached by a projectile. This height is the highest point in the projectile's trajectory.
Position and Velocity Analysis: Analyzing the horizontal and vertical components of the projectile's motion separately allows us to calculate its position and velocity at any given time. The horizontal motion is characterized by a constant velocity, while the vertical motion is affected by acceleration due to gravity.
Application: Understanding projectile motion is crucial in various fields such as physics, engineering, and sports. The ability to predict the path of a projectile helps in designing structures, aiming projectiles accurately, and optimizing performance in sports.
In conclusion, the calculations for the maximum height reached by a projectile fired at an angle of 53° with an initial velocity of 50 m/s and the position/velocity of the projectile 6 seconds later provide valuable insights into the dynamics of projectile motion. By applying the principles of kinematics and analyzing both horizontal and vertical components, we can understand and predict the motion of projectiles with precision.