A hunter aims directly at a target (on the same level) 38.0 m away. 
(b) At what angle should the bow be aimed so the target will be hit?

Verified step by step guidance

1. Identify the initial velocity (v_0) of the arrow. This information might be given, or you may need to estimate based on typical values for a bow.
2. Recognize that the arrow follows a parabolic trajectory due to gravity. The only force acting on the arrow after it is shot is gravity, which affects the vertical motion.
3. Use the projectile motion equation for horizontal displacement, x = v_0 \cos(\theta) \cdot t, where x is the horizontal distance, \theta is the angle of projection, and t is the time of flight.
4. Use the equation for vertical displacement, y = v_0 \sin(\theta) \cdot t - \frac{1}{2} g t^2, where y is the vertical displacement (which is zero since the target is at the same level as the bow) and g is the acceleration due to gravity (approximately 9.81 m/s^2). Set y to 0 to solve for t.
5. Substitute the expression for t from the vertical displacement equation into the horizontal displacement equation and solve for \theta, the angle at which the bow must be aimed.

Key Concepts

Projectile Motion

Projectile motion refers to the motion of an object that is launched into the air and is subject to gravitational forces. It can be analyzed in two dimensions: horizontal and vertical. The horizontal motion is uniform, while the vertical motion is influenced by gravity, leading to a parabolic trajectory. Understanding this concept is crucial for determining the angle of launch needed to hit a target at a specific distance.

Angle of Projection

The angle of projection is the angle at which an object is launched relative to the horizontal. This angle significantly affects the range and height of the projectile. For a given initial velocity, there is an optimal angle (usually 45 degrees) that maximizes the distance traveled. In this scenario, calculating the correct angle is essential to ensure the arrow reaches the target 38.0 m away.

Range Equation

The range equation in projectile motion relates the horizontal distance traveled by the projectile to its initial velocity and the angle of projection. It is given by the formula R = (v^2 * sin(2θ)) / g, where R is the range, v is the initial velocity, θ is the angle of projection, and g is the acceleration due to gravity. This equation is vital for determining the necessary angle to hit the target at a specified distance.