Physics Problem

Question

An experienced archer shot an arrow from the top of a fortress with a launch angle of +15.0° from the horizontal direction. The arrow moved at 43.3 m/s at the instant it was at the top of its trajectory, and landed at a point in the midst of a hostile army, after 3.62 seconds of flight in the air. Let us ignore air resistance for simplicity. (a) How far, as a straight line distance, is the landing point from the launch point? (b) What is the "landing" speed of the arrow?

Answer 1

For this projectile problem, we need to remember a_x= 0 (no air resistance) and a_y =-9.8m/s² (gravity)

Initial velocity v_i=√(v_x²+v_y²) but v_x = v_i cos 15^° = 43.3m/s is given. We can use this to find v_i.

The horizontal distance traveled is v_xt (both quantities given) = (43.3)(3.62)=156m

The landing speed is v_f=√(v_x²+v_yf²) where v_yf=v_yo+a_yt = (43.3)(sin 15^° /cos 15^°) -9.8(3.62) = 11.602-35.476

v_yf=28.126 v_x=43.3 so v_f=√[(-28.126)²+43.3²]=49.45 m/s

Answer 2

The speed of the arrow at the top of its flight (43.3 m/s) provides the (constant) horizontal component of the arrow's velocity. As long as the arrow is in flight, it moves horizontally at 43.3 m/s.

Using the launch angle and the horizontal velocity, we can determine the initial vertical velocity of the arrow. The tangent of the launch angle (15 degrees) is equal to the vertical component of the launch velocity divided by the horizontal component (tan = opposite/adjacent).

Using time of flight and horizontal speed, we can determine the range of the arrow - how far from the base of the fortress it strikes the ground. Since horizontal speed is constant, simply multiply speed by time.

We can use the initial vertical speed to determine the height of the fortress. Set vertical displacement equal to v_i t + 1/2 a t², with a = g = -9.8 m/s².

The straight line distance from launch point to landing point can be determined from horizontal and vertical displacements - use the Pythagorean Theorem.

For landing speed, we need first to determine the vertical component of the velocity at the time the arrow lands. Use v = v_i + at, again using a = g = 9.8. Then use Pythagorean Theorem, with x and y velocity components, to determine landing speed.