A gray kangaroo can bound across level ground with each jump carrying it 10 m from the takeoff point. Typically the kangaroo leaves the ground at a 20° angle. If this is so: What is its maximum height above the ground?

Verified step by step guidance

1. Step 1: Identify the given values in the problem. The kangaroo's jump covers a horizontal distance of 10 m, and the angle of takeoff is 20°. The goal is to find the maximum height above the ground. To do this, we need to analyze the vertical motion of the kangaroo using kinematic equations.
2. Step 2: Break the initial velocity into horizontal and vertical components. Let the initial velocity be \( v_0 \). The vertical component of the velocity is \( v_{0y} = v_0 \sin(\theta) \), and the horizontal component is \( v_{0x} = v_0 \cos(\theta) \), where \( \theta = 20° \).
3. Step 3: Use the horizontal motion to find the initial velocity \( v_0 \). The horizontal range \( R \) is given by \( R = \frac{v_0^2 \sin(2\theta)}{g} \), where \( g \) is the acceleration due to gravity (\( 9.8 \, \text{m/s}^2 \)). Rearrange this equation to solve for \( v_0 \).
4. Step 4: Once \( v_0 \) is determined, calculate the vertical component of the velocity \( v_{0y} \). The maximum height \( h_{\text{max}} \) is reached when the vertical velocity becomes zero. Use the kinematic equation \( v_y^2 = v_{0y}^2 - 2gh_{\text{max}} \), where \( v_y = 0 \) at the maximum height. Rearrange this equation to solve for \( h_{\text{max}} \): \( h_{\text{max}} = \frac{v_{0y}^2}{2g} \).
5. Step 5: Substitute \( v_{0y} = v_0 \sin(\theta) \) into the equation for \( h_{\text{max}} \). This gives \( h_{\text{max}} = \frac{(v_0 \sin(\theta))^2}{2g} \). Use the value of \( v_0 \) obtained earlier and the given angle \( \theta = 20° \) to complete the calculation.

Key Concepts

Projectile Motion

Projectile motion refers to the motion of an object that is launched into the air and is influenced only by the force of gravity and its initial velocity. In this case, the kangaroo's jump can be analyzed as a projectile, where its trajectory is determined by the angle of launch and the initial speed. Understanding the components of projectile motion, including horizontal and vertical motions, is essential for calculating maximum height.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are crucial for resolving the initial velocity of the kangaroo's jump into its vertical and horizontal components. The angle of launch (20°) allows us to use these functions to determine how much of the initial velocity contributes to the upward motion, which directly affects the maximum height reached during the jump.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration, such as gravity. For the kangaroo's jump, we can use these equations to relate the initial vertical velocity, the maximum height, and the acceleration due to gravity. Specifically, the equation that relates maximum height to initial velocity and acceleration will help us find the height above the ground.