Chapter 6, Problem 90RE

Solution

To find:When Gretta throws the dart at an angle of $50°$ with the horizontal from a height of 5 ft above the ground with an initial velocity of $20\text{ft}/\sec$ , then the dart will hit the target or not?

No

Given information:

The distance from Gretta and Lois to the target is 20 ft. Radius of the circular target is 18 inches.

Formula used:

The path of an object in projectile motion is modelled by the parametric equations $x=\left(v_0\cos \theta \right)t$ and $y=-16t^2+\left(v_0\sin \theta \right)t+y_0$ , where $y_0$ is the point above the ground from where the object is thrown, $v_0$ is the initial speed in $\text{ft}/\sec$ at an angle $\theta$ with horizontal.

Calculation:

It is known that the path of an object in projectile motion is modelled by the parametric equations $x=\left(v_0\cos \theta \right)t$ and $y=-16t^2+\left(v_0\sin \theta \right)t+y_0$ , where $y_0$ is the point above the ground from where the object is thrown, $v_0$ is the initial speed in $\text{ft}/\sec$ at an angle $\theta$ with horizontal.

In the given situation, $v_0=20\text{ft}/\sec$ , $\theta =50°$ , and $y_0=5\text{ft}$ .

Substitute these values into the parametric equations $x=\left(v_0\cos \theta \right)t$ and $y=-16t^2+\left(v_0\sin \theta \right)t+y_0$ .

  $x=\left(20\cos 50°\right)t$ and $y=-16t^2+\left(20\sin 50°\right)t+5$

Evaluate the value of $t$ when $x=20\text{ft}$ .

  $\begin{array}{c}20=\left(20\cos 50°\right)t\\ t=\frac{20}{20\cos 50°}\\ t\approx 1.556\sec \end{array}$

Substitute $t\approx 1.556\sec$ into $y=-16t^2+\left(20\sin 50°\right)t+5$ and evaluate the value of $y$ .

  $\begin{array}{l}y=-16{\left(1.556\right)}^2+\left(20\sin 50°\right)\left(1.556\right)+5\\ y\approx -9.9\text{ft}\end{array}$

Here, the negative sign implies that the dart hit the ground before reaching the target which is 20 ft away from the point the dart was thrown. So, the dart didn’t hit the target. Therefore, the answer is “No”.