Question 4.3: A Bull’s-Eye Every Time In a popular lecture demonstration, ...
A Bull’s-Eye Every Time
In a popular lecture demonstration, a projectile is aimed directly at a target and fired in such a way that the projectile leaves the gun at the same time the target is dropped from rest. Show that the projectile hits the falling target.
Conceptualize We conceptualize the problem by studying Figure 4.13a. Notice that the problem does not ask for numerical values. The expected result must involve an algebraic argument.
Categorize Because both objects are subject only to gravity, we categorize this problem as one involving two objects in free fall, the target moving in one dimension and the projectile moving in two. The target T is modeled as a particle under constant acceleration in one dimension. The projectile P is modeled as a particle under constant acceleration in the y direction and a particle under constant velocity in the x direction.
Analyze Figure 4.13b shows that the initial y coordinate y_{iT} of the target is x_T tan \theta_i and its initial velocity is zero. It falls with acceleration a_y =-g.
Write an expression for the y coordinate of the target at any moment after release, noting that its initial velocity is zero:
(1) y_{T}=y_{i T}+(0) t-\frac{1}{2} g t^2=x_{T} \tan \theta_i-\frac{1}{2} g t^2
Write an expression for the y coordinate of the projectile at any moment:
(2) y_{P}=y_{i P}+v_{y i P} t-\frac{1}{2} g t^2=0+\left(v_{i P} \sin \theta_i\right) t-\frac{1}{2} g t^2=\left(v_{i P} \sin \theta_i\right) t-\frac{1}{2} g t^2
Write an expression for the x coordinate of the projectile at any moment:
x_{P}=x_{i P}+v_{x i P} t=0+\left(v_{i P} \cos \theta_i\right) t=\left(v_{i P} \cos \theta_i\right) tSolve this expression for time as a function of the horizontal position of the projectile:
t=\frac{x_{P}}{v_{i P} \cos \theta_i}Substitute this expression into Equation (2):
(3) y_{P}=\left(v_{i P} \sin \theta_i\right)\left(\frac{x_{P}}{v_{i P} \cos \theta_i}\right)-\frac{1}{2} g t^2=x_{P} \tan \theta_i-\frac{1}{2} g t^2
Finalize Compare Equations (1) and (3). We see that when the x coordinates of the projectile and target are the same—that is, when x_T = x_p—their y coordinates given by Equations (1) and (3) are the same and a collision results.
