Question

Use the model for projectile motion, assuming there is no air resistance. A projectile is fired from ground level at an angle of \(12^{\circ}\) with the horizontal. The projectile is to have a range of 150 feet. Find the minimum initial velocity necessary.

Short answer

The minimum initial velocity is obtained by inserting the values for range, gravity, and angle (in radians) into the formula and solving for velocity. The exact calculation will depend on the given angle and range.

Step-by-step solution

Convert the Initial Angle from Degrees to Radians

The initial angle \(\theta\) of the projectile is given as \(12^\circ\). However, the formula for the range requires the angle to be in radians. To convert from degrees to radians, multiply the angle in degrees by \(\frac{{\pi}}{{180}}\). In this case, \(\theta = 12^\circ \times \(\frac{{\pi}}{{180}}\).\)

Insert the Values into the Formula and Solve for Initial Velocity

Insert the range R (150 feet), the converted angle value, and the acceleration due to gravity g (32.2 ft/s²) into the formula. The formula becomes \(150 = \frac{{v^2 \sin(2 \times \theta)}}{32.2}\). Solving for v, the equation becomes \(v = \sqrt{\frac{{150 \times 32.2}}{\sin(2 \times \theta)}}\)

Calculate the Initial Velocity

Insert the value of \(\theta\) from step 1 into the formula obtained in step 2. You should reach a result for the initial velocity necessary for the projectile to have a range of 150 feet