A projectile has a range of 50m and reaches a maximum height of 10m. Calculate the angle at which the projectile is fired.
H/ R = tan θ / 4
H = 10m.
R = 50m.
Tanθ = 4 × 10/50 = 0.8
θ = Arctan 0.8
To calculate the angle at which the projectile is fired, we can use the formula for the range of a projectile:
Range = (v^2 * sin(2θ)) / g
where v is the initial velocity of the projectile, θ is the angle at which it is fired, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Given that the range is 50m, we can rearrange the formula to solve for the angle θ:
θ = arcsin((g * Range) / v^2)
However, we don't have the value of the initial velocity, v, in this problem.
To solve for v, we can use the formula for the maximum height of a projectile:
Height = (v^2 * sin^2(θ)) / (2 * g)
Given that the maximum height is 10m, we can rearrange this formula to solve for v^2:
v^2 = (Height * 2 * g) / sin^2(θ)
Now, we can substitute this expression for v^2 into the formula for the angle:
θ = arcsin((g * Range) / ((Height * 2 * g) / sin^2(θ)))
To simplify this equation, let's cancel out common factors:
θ = arcsin((2 * Range) / (Height * sin^2(θ)))
Now we have an equation with only one unknown, θ. However, solving this equation analytically is quite complex as it involves solving a trigonometric equation. Therefore, it's best to use numerical methods or a calculator to find the value of θ that satisfies the equation.
Using a scientific calculator, you can follow these steps:
1. Substitute the given values: Range = 50m, Height = 10m.
2. Choose a starting value for θ (e.g., 30 degrees) and calculate the left-hand side of the equation.
3. Adjust the value of θ iteratively, plugging it into the equation and calculating the left-hand side until you get a close approximation of the right-hand side.
4. Once you find the value of θ that satisfies the equation, that will be the angle at which the projectile is fired.
To calculate the angle at which the projectile is fired, we can use the following formula:
Range = (Initial velocity^2 * sin(2*θ)) / g
Where:
- Range is the horizontal distance traveled by the projectile
- Initial velocity is the magnitude of the projectile's velocity when fired
- θ is the angle at which the projectile is fired
- g is the gravitational acceleration (approximately 9.8 m/s^2)
In this case, we have:
Range = 50m
Maximum height = 10m
To find the angle, we need to rearrange the formula and solve for θ.
θ = arcsin ((Range * g) / (Initial velocity^2 * 2))
We do not have the value for the initial velocity, so we cannot directly calculate the angle without further information.