Question

A soccer ball with a mass of \(442 \mathrm{~g}\) bounces off the crossbar of a goal and is deflected upward at an angle of \(58.0^{\circ}\) with respect to horizontal. Immediately after the deflection, the kinetic energy of the ball is \(49.5 \mathrm{~J} .\) What are the vertical and horizontal components of the ball's momentum immediately after striking the crossbar?

Short answer

Answer: The vertical and horizontal components of the ball's momentum are approximately 3.32 kg⋅m/s and 5.33 kg⋅m/s, respectively.

Step-by-step solution

Calculate the initial velocity of the ball

First, we'll use the given kinetic energy to find the initial velocity of the ball after the deflection. The formula to calculate kinetic energy is: \(KE = \frac{1}{2}mv^2\) We know that the kinetic energy (\(KE\)) is \(49.5 J\), the mass (\(m\)) is \(442 g = 0.442 kg\) and we need to find the velocity (\(v\)). We can rewrite the formula to solve for \(v\): \(v = \sqrt{\frac{2 \times KE}{m}}\)

Plug in the values and compute the initial velocity

Now, we plug in the values of \(KE\) and \(m\) into the formula: \(v = \sqrt{\frac{2 \times 49.5 J}{0.442 kg}}\) \(v \approx 14.05\,\text{m/s}\)

Calculate the vertical and horizontal components of the velocity

Now that we have the initial velocity, we can compute the vertical and horizontal components of the velocity using the deflection angle (\(58.0^\circ\)). We use trigonometry to find these components: \(v_x = v\cos(58.0^\circ)\) \(v_y = v\sin(58.0^\circ)\)

Plug in the values and compute the velocity components

We'll now plug in the values of \(v\) and the angle to compute \(v_x\) and \(v_y\): \(v_x = 14.05\,\text{m/s} \times \cos(58.0^\circ) \approx 7.52\,\text{m/s}\) \(v_y = 14.05\,\text{m/s} \times \sin(58.0^\circ) \approx 12.05\,\text{m/s}\)

Calculate the vertical and horizontal components of the momentum

Finally, we can find the vertical and horizontal components of the momentum by multiplying the respective components of the velocity by the mass of the ball: \(p_x = m \times v_x\) \(p_y = m \times v_y\)

Plug in the values and compute the momentum components

Now, we'll plug in the values of \(m\), \(v_x\), and \(v_y\) to compute \(p_x\) and \(p_y\): \(p_x = 0.442 kg \times 7.52\,\text{m/s} \approx 3.32\,\text{kg}\cdot\text{m/s}\) \(p_y = 0.442 kg \times 12.05\,\text{m/s} \approx 5.33\,\text{kg}\cdot\text{m/s}\) The vertical and horizontal components of the ball's momentum immediately after striking the crossbar are approximately \(3.32\,\text{kg}\cdot\text{m/s}\) and \(5.33\,\text{kg}\cdot\text{m/s}\), respectively.

Key concepts

Kinetic Energy

Kinetic energy is a measure of the energy that an object possesses due to its motion. It is defined by the formula: \(KE = \frac{1}{2}mv^2\), where \(m\) is the mass of the object and \(v\) is its velocity. This energy determines how much work an object can do due to its movement. In our example, a soccer ball with a mass of 442 grams has a kinetic energy of 49.5 Joules after bouncing off a crossbar, which is measured in joules (J). To find velocity from kinetic energy, we rearrange the formula to: \(v = \sqrt{\frac{2 \times KE}{m}}\). This allows us to determine the ball's speed after the deflection. Understanding kinetic energy helps in linking motion to energy which is crucial in physics, as it connects mass and velocity to energy output.

Velocity Components

When an object moves in a plane, its velocity can be split into two components: horizontal and vertical. This separation is essential in analyzing the object's overall motion. By knowing the velocity angle, we can determine these components through trigonometry.

• Horizontal component \(v_x = v \cos(\theta)\)
• Vertical component \(v_y = v \sin(\theta)\)

For the soccer ball, given its deflection angle of \(58.0^\circ\), the horizontal velocity component \(v_x\) becomes \(14.05 \times \cos(58.0^\circ)\), and the vertical component \(v_y\) becomes \(14.05 \times \sin(58.0^\circ)\).These components describe how fast the ball travels in each direction immediately after striking the crossbar.

Trigonometry

Trigonometry is a branch of mathematics that explores the relationships between the angles and sides of triangles. It is especially useful for solving problems involving angles and distances.To determine velocity components, we employ two functions:

• Cosine (\(\cos\)
- relates the adjacent side to the hypotenuse in a right triangle.
• Sine (\(\sin\)
- relates the opposite side to the hypotenuse in a right triangle.

With a deflection angle of \(58.0^\circ\), cosine gives the horizontal direction component, and sine gives the vertical direction component. Trigonometry allows us to decompose complex motion into simpler, directional components, which are easier to analyze and compute.

Deflection Angle

A deflection angle describes the direction change of an object's path after it interacts with another surface. In our case, the angle is \(58.0^\circ\) above the horizontal after the soccer ball hits the crossbar. This angle is crucial as it influences the calculation of velocity components.The deflection angle helps to determine how the original kinetic energy distributes between the horizontal and vertical movements of the ball.When using trigonometry functions like cosine and sine, the angle ensures precise calculations of these components. Consequently, the deflection angle dictates the new path's steepness and speed, understanding these concepts aids in predicting an object's trajectory post-impact.