Question

Use the model for projectile motion, assuming there is no air resistance. Find the angle at which an object must be thrown to obtain (a) the maximum range and (b) the maximum height.

Short answer

The angle to obtain maximum range is 45 degrees, while the angle to get the maximum height is 90 degrees.

Step-by-step solution

Identify the model for maximum range

The formula to find the maximum range (R) in the case of projectile motion is given by \[ R = \frac{v^2}{g} \sin 2\theta \] where v is the initial velocity of the object, g is the acceleration because of gravity, and θ is the angle. This range will be maximum when the value of \(\sin 2\theta\) is maximum, which is 1. The maximum value of the sine function occurs at an angle of \(\frac{\pi}{2}\) radians, or 90 degrees. So, set \(2\theta = \frac{\pi}{2}\) and solve for \(\theta\).

Calculate angle for maximum range

For maximum range, \[2\theta = \frac{\pi}{2}\] Solving for \(\theta\) we get: \[\theta = \frac{\pi}{4}\] or 45 degrees.

Identify the model for maximum height

The equation to compute the maximum height (H) during projectile motion is \[ H = \frac{v^2 \sin^2 \theta}{2g} \] In this case, height will be maximum when the value of \(\sin^2 \theta\) is maximum, which is again 1. The maximum value of the sine function occurs at an angle of 90 degrees. So, the angle for maximum height is 90 degrees.