Question

1. the following projectiles are launched on horizontal ground with the same initial speed. if two or three situations have the same answer, put the letters in the same blank space. air resistance is negligible.

(a) rank the situations from least to greatest in terms of time the projectile is in the air.
least 1__ 2 3 4__ greatest
or, all of the projectiles are the same ____
(b) rank the situations from least to greatest in terms of the horizontal distance the projectile travels while in the air.
least 1__ 2 3 4__ greatest
or, all of the projectiles are the same ____
(c) rank the situations from least to greatest in terms of the vertical distance the projectile travels while in the air.
least 1__ 2 3 4__ greatest
or, all of the projectiles are the same ____
(d) rank the situations from least to greatest in terms of the acceleration of the projectile while it is in the air.
least 1__ 2 3 4__ greatest
or, all of the projectiles are the same

Explanation:

Step1: Analyze time - of - flight formula

The time - of - flight \(T\) of a projectile launched with initial speed \(v_0\) at an angle \(\theta\) from the horizontal on a horizontal ground (\(y = y_0\)) is given by \(T=\frac{2v_0\sin\theta}{g}\), where \(g\) is the acceleration due to gravity. For situation A, \(\theta_A\approx15^{\circ}\), for B, \(\theta_B\approx30^{\circ}\), for C, \(\theta_C\approx45^{\circ}\), and for D, \(\theta_D = 90^{\circ}\). Since \(\sin15^{\circ}<\sin30^{\circ}<\sin45^{\circ}<\sin90^{\circ}\), the time of flight is \(T_A < T_B < T_C < T_D\).

Step2: Analyze horizontal - range formula

The horizontal range \(R\) of a projectile is \(R=\frac{v_0^{2}\sin2\theta}{g}\). \(\sin2\theta_A=\sin30^{\circ}=0.5\), \(\sin2\theta_B=\sin60^{\circ}=\frac{\sqrt{3}}{2}\), \(\sin2\theta_C=\sin90^{\circ}=1\), \(\sin2\theta_D=\sin180^{\circ}=0\). So \(R_D < R_A < R_B < R_C\).

Step3: Analyze vertical - distance

The vertical distance \(y - y_0=v_{0y}t-\frac{1}{2}gt^{2}\). The maximum vertical displacement \(y - y_0\) occurs at \(t = \frac{v_{0y}}{g}\). For a projectile launched with initial vertical velocity \(v_{0y}=v_0\sin\theta\), the maximum height \(H=\frac{(v_0\sin\theta)^{2}}{2g}\). Since \(\sin15^{\circ}<\sin30^{\circ}<\sin45^{\circ}<\sin90^{\circ}\), the vertical distance is \(A < B < C < D\).

Step4: Analyze acceleration

The acceleration of a projectile in free - fall (neglecting air resistance) is always \(a = g\) (downward) for all cases. So the acceleration is the same for all projectiles.

Answer:

(a) A, B, C, D
(b) D, A, B, C
(c) A, B, C, D
(d) all of the projectiles are the same