Question

For a Science Olympiad competition, a group of middle school. students build a trebuchet that can fire a tennis ball from a height of \(1.55 \mathrm{~m}\) with a velocity of \(10.5 \mathrm{~m} / \mathrm{s}\) and a launch angle of \(35.0^{\circ}\) above the horizontal. What horizontal distance will the tennis ball cover before it hits the ground?

Short answer

Answer: The tennis ball will cover a horizontal distance of approximately 5.13 meters before hitting the ground.

Step-by-step solution

Calculate the initial horizontal and vertical velocities

To find the initial horizontal and vertical velocities, we'll use trigonometry. The initial velocity is given as 10.5 m/s at a launch angle of 35 degrees. Using the sine function for vertical velocity and cosine function for horizontal velocity: Horizontal velocity: \(v_{0x} = v_0 \times \cos(\theta) = 10.5 \times \cos(35^\circ) \approx 8.61 \mathrm{~m/s}\) Vertical velocity: \(v_{0y} = v_0 \times \sin(\theta) = 10.5 \times \sin(35^\circ) \approx 6.03 \mathrm{~m/s}\)

Find the time in air

To find the time the tennis ball spends in the air, we need to consider its motion in the vertical direction. We will use the following equation of motion: \(h = h_0 + v_{0y}t - \frac{1}{2}gt^2\) Where \(h\) is the final height (0 m since the ball hits the ground), \(h_0\) is the initial height (1.55 m), \(v_{0y}\) is the initial vertical velocity, \(t\) is the time, and \(g\) is the acceleration due to gravity (9.8 m/s²). Plugging in the given values and solving for the time: \(0 = 1.55 + 6.03t - 4.900t²\) This is a quadratic equation of the form \(4.900t^2 - 6.03t + 1.55 = 0\). To find the time, we will use the quadratic formula: \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) Where \(a = 4.9\), \(b = -6.03\), and \(c = 1.55\). Solving for \(t\), we will have two solutions. We'll take the positive one since time cannot be negative. \(t \approx 0.596 \mathrm{~s}\)

Calculate the horizontal distance

Now that we have the time the ball is in the air, we can find the horizontal distance traveled. The horizontal motion is not affected by gravity, so we can use the following equation: \(d_x = v_{0x}t\) Plugging in the values for \(v_{0x}\) and \(t\): \(d_x = 8.61 \mathrm{~m/s} \times 0.596 \mathrm{~s} \approx 5.13 \mathrm{~m}\) The tennis ball will cover a horizontal distance of approximately 5.13 meters before it hits the ground.

Key concepts

Trebuchet Physics

Understanding trebuchet physics is crucial for grasping how these medieval siege engines, as well as their modern-day science project counterparts, work. A trebuchet leverages the principles of mechanical advantage and conservation of energy to launch projectiles over long distances. The main components of a trebuchet include a long beam, a counterweight, a sling, and a pivot.

The counterweight, when released, uses gravity to provide energy that is transferred into the beam and then to the projectile in the sling. As the beam rotates around the pivot, the sling holding the projectile moves in an increasing arc, and at the optimal point, the projectile is released. The energy transfer involves potential energy from the counterweight converted into kinetic energy in the projectile.

When calculating the physics behind a trebuchet's launch, points to consider include the mass of the counterweight, the length of the beam and sling, and the angle of release. These factors affect the projectile's velocity and trajectory, ultimately determining how far it will travel.

Trigonometry in Physics

Trigonometry, the study of relationships between the angles and sides of triangles, is an integral tool in the field of physics, particularly when analyzing projectile motion. Trigonometric functions such as sine and cosine provide a means to resolve a force or velocity into its perpendicular components. For instance, in our exercise involving the tennis ball launched from a trebuchet at an angle, we used the cosine function to calculate the horizontal component of the velocity and the sine function to calculate the vertical component.

These trigonometric calculations allow us to treat the horizontal and vertical motions independently, which simplifies the analysis of projectile motion. The angle of launch, measured from the horizontal, plays a significant role in determining the initial velocity components, which in turn affects how high and how far the projectile will go. These practices show how trigonometry continues to be a fundamental aspect of understanding and solving problems in physics.

Equations of Motion

The equations of motion are a set of formulas that describe the mathematical relationship between an object's displacement, velocity, acceleration, and time. They are essential for predicting the future state of moving objects, provided their initial state is known. In projectile motion, like the trajectory of the tennis ball from the trebuchet, two critical equations are often employed.

Firstly, the equation for the vertical motion, considering acceleration due to gravity, is \( h = h_0 + v_{0y}t - \frac{1}{2}gt^2 \). This equation, which incorporates the initial height (\( h_0 \)) and the initial vertical velocity (\( v_{0y} \)), allows us to solve for the time the object stays in the air before landing at height \( h \).

Secondly, for horizontal motion, where there is no acceleration (as gravity does not affect it), the distance covered is given by the equation \( d_x = v_{0x}t \), with \( v_{0x} \) representing the initial horizontal velocity and \( t \) being the time derived from the vertical motion calculation. Together, these equations of motion enable us to forecast travel distance and duration, illustrating the predictability of projectile trajectories under constant acceleration.