Question

A soccer ball with mass \(0.265 \mathrm{~kg}\) is initially at rest and is kicked at an angle of \(20.8^{\circ}\) with respect to the horizontal. The soccer ball travels a horizontal distance of \(52.8 \mathrm{~m}\) after it is kicked. What is the impulse received by the soccer ball during the kick? Assume there is no air resistance.

Short answer

The soccer ball receives an impulse of approximately 6.75 kg⋅m/s during the kick.

Step-by-step solution

Determine the initial velocity

To find the impulse, we first need to know the initial velocity of the soccer ball. Since there is no air resistance, we are dealing with a projectile motion problem. The horizontal distance traveled by the soccer ball can be described by the following equation: $$x = v_0 \cos\theta \cdot t$$ Where \(x\) is the horizontal distance, \(v_0\) is the initial velocity, \(\theta\) is the angle at which the ball was kicked, and \(t\) is the time of flight. We can also find an expression for the time of flight using the vertical motion equation: $$y = v_0 \sin\theta \cdot t - \frac{1}{2}gt^2$$ Since the soccer ball lands on the ground, we have \(y=0\). By solving this equation for \(t\) and substituting it back into the horizontal distance equation, we can get an expression for the initial velocity.

Calculate the time of flight

First, let's solve the vertical motion equation for \(t\): $$0 = v_0 \sin\theta \cdot t - \frac{1}{2}gt^2$$ The quadratic formula gives us: $$t = \frac{2v_0 \sin\theta}{g}$$

Calculate the initial velocity

Now, we substitute the expression for \(t\) from Step 2 into the horizontal distance equation: $$x = v_0 \cos\theta \cdot \frac{2v_0 \sin\theta}{g}$$ $$52.8 \mathrm{~m} = v_0^2 \sin20.8^{\circ} \cos20.8^{\circ} \cdot \frac{2}{9.81 \mathrm{~m/s^2}}$$ Solving for the initial velocity, we get: $$v_0 = \sqrt{\frac{52.8 \mathrm{~m} \cdot 9.81 \mathrm{~m/s^2}}{2 \sin20.8^{\circ} \cos20.8^{\circ}}} \approx 25.46 \mathrm{~m/s}$$

Calculate the impulse

Now that we have the initial velocity, we can calculate the impulse, which is given by the change in momentum: $$I = m \Delta v = m(v_f - v_i)$$ The soccer ball was initially at rest, so \(v_i = 0 \mathrm{~m/s}\). The final velocity is the initial velocity after the kick, which is \(v_f = 25.46 \mathrm{~m/s}\). The mass of the soccer ball is \(0.265 \mathrm{~kg}\). Therefore, the impulse is: $$I = (0.265 \mathrm{~kg}) (25.46 \mathrm{~m/s} - 0 \mathrm{~m/s}) \approx 6.75 \mathrm{~kg \cdot m/s}$$ So, the impulse received by the soccer ball during the kick is approximately \(6.75 \mathrm{~kg \cdot m/s}\).

Key concepts

Projectile Motion

Projectile motion refers to the two-dimensional motion of an object that is launched into the air and subject to only gravitational acceleration. A key characteristic of projectile motion is that the horizontal and vertical motions are independent of each other, except for the time they take. This means that the horizontal velocity remains constant since there are no additional forces in the horizontal direction (assuming no air resistance), while the vertical velocity changes because of gravity.

When an object is launched at an angle, the initial velocity vector (\(v_0\)) has both horizontal and vertical components. These components can be found using trigonometry: the horizontal component is \(v_0 \cos(\theta)\), and the vertical component is \(v_0 \sin(\theta)\), where \(\theta\) is the launch angle. The object will follow a parabolic trajectory and eventually return to the same vertical level it was launched from, which in the case of the soccer ball is when it touches the ground again (\(y = 0\)).

Initial Velocity Calculation

Determining the initial velocity is pivotal for solving projectile motion problems. The initial velocity can be calculated if you know the distance traveled, the angle of launch, and the acceleration due to gravity (\(g\)). The previous explanation provides a method for determining the initial velocity of a soccer ball in such a scenario by using the horizontal range of the projectile and the time of flight.

In situations where air resistance can be neglected, calculating the initial velocity involves first finding the time of flight using the vertical motion equation. You can use the derived time of flight to substitute into the horizontal distance formula. This process eventually allows you to solve for the initial velocity. The calculation reveals the speed at which the projectile must move to cover the desired distance, considering the influence of gravity on its vertical motion.

Momentum Change

Momentum, in physics, is the quantity of motion of a moving body and is calculated as the product of mass and velocity (\(p = mv\)). The change in momentum, or the impulse, occurs whenever a force is applied over a time interval. Impulse has the same units as momentum (\(\mathrm{kg \cdot m/s}\)) and is given by the change in momentum of an object.

The formula for impulse (\(I\)) is essentially the mass of the object multiplied by the change in velocity (\(\Delta v\)), which is \(I = m(v_f - v_i)\). It's significant to remember that momentum is a vector quantity, so direction is essential: an impulse can alter either the magnitude or the direction of the momentum, or both. In the scenario of the soccer ball, the ball initially at rest receives an impulse that corresponds to the ball's subsequent motion, determined by the final velocity right after the kick.