When projectile angle is 45°what is the relation between maximum height and range
R = 2h
When a projectile is launched at an angle of 45 degrees to the horizontal, the maximum height it achieves is equal to half of its total range.
To understand this relation, let's consider the motion of the projectile. The path of the projectile can be divided into two components: horizontal (x) and vertical (y).
At an angle of 45 degrees, the initial velocity of the projectile is evenly split between the horizontal and vertical components. This means that the initial velocity in the y-direction is equal to the initial velocity in the x-direction.
As the projectile moves upward in the y-direction, it decelerates due to the pull of gravity until it reaches its maximum height. At the highest point of its trajectory, the vertical velocity of the projectile becomes zero. This happens because the gravitational force acting in the downward direction cancels out the projectile's upward motion.
On the way down, the projectile accelerates downward due to the gravitational force, eventually reaching the same speed it had initially but in the opposite direction. The time taken to reach the maximum height is equal to the time taken to return to the same height after reaching the maximum height.
Now, let's consider the range of the projectile. The range is defined as the horizontal distance traveled by the projectile before hitting the ground. At the maximum height, the y-component of the velocity is zero, but the x-component of the velocity is still nonzero and equal to the initial velocity in the x-direction. This means that the projectile continues to move horizontally.
The time taken for the projectile to reach the maximum height is the same as the time taken for it to fall back to the ground. Therefore, half of the total time of flight is spent in the ascending phase of the motion, and the other half is spent in the descending phase of the motion.
Since the time of flight is double the time taken to reach the maximum height, the time taken to reach the maximum height is therefore equal to half of the total time of flight. As a result, the projectile spends an equal amount of time in the ascending and descending phases of its motion.
Since the horizontal velocity remains constant throughout the motion, the range is determined by the time of flight. The total time of flight is thus twice the time taken to reach the maximum height.
Combining all of this information, we can conclude that when a projectile is launched at an angle of 45 degrees, the maximum height it achieves is equal to half of its total range.
To determine the relationship between the maximum height and range of a projectile when launched at an angle of 45°, we need to consider the laws of projectile motion.
Projectile motion is the motion of an object projected into the air and moving under the influence of only gravity, neglecting air resistance. When launched at an angle of 45°, the maximum height and the range of the projectile will be related.
The maximum height (H) of a projectile can be determined using the following formula:
H = (v^2 * sin^2θ) / (2 * g)
Where:
H = maximum height,
v = initial velocity of the projectile,
θ = launch angle, and
g = acceleration due to gravity (approximately 9.8 m/s^2).
The range (R) of a projectile, which is the horizontal distance traveled, can be calculated using this formula:
R = (v^2 * sin(2θ)) / g
Therefore, when the projectile is launched at an angle of 45°, the launch angle (θ) in both formulas will be the same (45°). This results in a simplified relationship between the maximum height and range of the projectile.
By substituting θ = 45° into the formulas, we get:
H = (v^2 * sin^2(45°)) / (2 * g)
R = (v^2 * sin(2 * 45°)) / g
Since sin(45°) and sin(2 * 45°) both equal 1/√2, we can simplify the equations further:
H = (v^2 / (2 * g)
R = (v^2 / g)
From these simplified equations, we can observe that the maximum height (H) is one-half of the range (R) when the projectile is launched at an angle of 45°. In other words, the maximum height reached by the projectile is equal to half of the horizontal distance it travels.