Question

A potato cannon is used to launch a potato on a frozen lake, as shown in the figure. The mass of the cannon, \(m_{c}\), is \(10.0 \mathrm{~kg}\), and the mass of the potato, \(m_{\mathrm{p}^{\prime}}\) is \(0.850 \mathrm{~kg}\). The cannon's spring (with spring constant \(k_{c}=7.06 \cdot 10^{3} \mathrm{~N} / \mathrm{m}\) ) is compressed \(2.00 \mathrm{~m}\). Prior to launching the potato, the cannon is at rest. The potato leaves the cannon's muzzle moving horizontally to the right at a speed of \(v_{p}=175 \mathrm{~m} / \mathrm{s}\). Neglect the effects of the potato spinning. Assume there is no friction between the cannon and the lake's ice or between the cannon barrel and the potato. a) What are the direction and magnitude of the cannon's velocity, \(v_{c}\) after the potato leaves the muzzle? b) What is the total mechanical energy (potential and kinetic) of the potato/cannon system before and after the firing of the potato?

Short answer

Answer: The direction of the cannon's velocity is opposite to the direction of the potato's velocity. The magnitude of the velocity can be calculated using the formula \(v_c = \dfrac{-p_p}{m_c}\), where \(p_p\) is the final momentum of the potato and \(m_c\) is the mass of the cannon. The total mechanical energy of the system before firing is given by the initial potential energy of the compressed spring, while the total mechanical energy after firing is the sum of the final kinetic energies of the potato and the cannon. These can be computed using the formulas provided in the solution above.

Step-by-step solution

Calculate the initial velocity of the potato cannon system

Since the potato cannon system is initially at rest, the initial total momentum of the system is 0.

Calculate the final momentum of the potato

Using the given mass and velocity of the potato, we can compute its final momentum. \(m_p = 0.850\,\text{kg}\) \(v_p = 175\,\text{m/s}\) \(p_p = m_p \cdot v_p\)

Calculate the final momentum of the cannon

According to the conservation of momentum in an isolated system, the total momentum is conserved. Since the initial momentum was 0, the final momentum of the cannon will be equal in magnitude but opposite in direction to the final momentum of the potato. \(p_c = m_c \cdot v_c = -p_p\)

Calculate the velocity of the cannon

Now we can solve for the velocity of the cannon by dividing its final momentum by its mass. \(m_c = 10.0\,\text{kg}\) \(\Rightarrow v_c = \dfrac{-p_p}{m_c}\) The negative sign indicates that the direction of the cannon's velocity is opposite to the direction of the potato's velocity. #b) Determine the total mechanical energy before and after the firing#

Calculate the initial spring potential energy

At the starting position, the spring is compressed 2.00 meters, so its potential energy can be computed using the following formula: \(E_{p, i} = \dfrac{1}{2} k_c x^2\) where \(k_c = 7.06 \cdot 10^{3}\,\text{N/m}\) and \(x = 2.00\,\text{m}\)

Calculate the initial kinetic energy

Since the system is initially at rest, the initial kinetic energy of both the potato and the cannon is zero.

Calculate the initial total mechanical energy

Adding the initial potential energy and initial kinetic energy, we can get the initial total mechanical energy of the system: \(E_{i} = E_{p, i} + 0\)

Calculate the final kinetic energy of the potato and the cannon

Using their final velocities and masses, we can calculate the final kinetic energy of both the potato and the cannon using the formula: \(E_{k,p} = \dfrac{1}{2} m_p v_p^2\) \(E_{k,c} = \dfrac{1}{2} m_c v_c^2\)

Calculate the final total mechanical energy

Adding the final kinetic energies of both the potato and the cannon, we can compute the final total mechanical energy of the system: \(E_{f} = E_{k,p} + E_{k,c}\)

Key concepts

Momentum Conservation

Momentum conservation is a pivotal concept in physics, particularly when analyzing collisions or the movement of objects. In the context of our potato cannon problem, we adhere to the principle that the momentum of the isolated system (cannon plus potato) before firing must be equal to the momentum after the potato is launched, assuming no external forces act on the system.

To put it simply, if the total momentum before firing is zero (as the system is at rest), then the total momentum after firing must also be zero. This means that the potato's forward momentum is exactly counterbalanced by the backward momentum of the cannon. This concept not only helps us determine the direction and magnitude of the cannon's recoil velocity but also reinforces the fundamental law of momentum conservation in an isolated system.

Mechanical Energy

Mechanical energy is the combined energy of motion and position of an object. It is usually categorized as kinetic energy, due to motion, and potential energy, stored energy associated with an object's position or configuration. To examine the mechanical energy before and after firing the potato cannon, one must consider both types of energy. Initially, the mechanical energy is purely potential, stored in the compressed spring of the cannon. After firing, this potential energy is converted into the kinetic energy of the moving potato and the recoiling cannon.

Understanding the transformation of energy in this scenario illustrates the conservation of mechanical energy principle and provides insights into the energy efficiency of the system. By comparing the initial and final mechanical energies, students can explore the concepts of energy transfer and transformation.

Kinetic Energy

Kinetic energy speaks to the energy an object possesses due to its motion, expressed mathematically as \(E_k = \frac{1}{2} m v^2\), where \(m\) is the object’s mass and \(v\) its velocity. In our exercise, both the potato and the cannon acquire kinetic energy as they move post-launch. The key takeaway here is that, although kinetic energy depends on both mass and velocity, because the potato has a much higher velocity compared to the cannon (despite its lower mass), its kinetic energy will be significantly high. By calculating the kinetic energies for both the potato and the cannon after the potato has been fired, we can assess the distribution of energy between the two objects and gain further appreciation for the role of mass and velocity in kinetic energy.

Potential Energy

Potential energy, often denoted as \(E_p\), is the stored energy in an object or system due to its position or arrangement. In terms of the potato cannon, the spring inside possesses potential energy when compressed, which is calculated with the formula \(E_p = \frac{1}{2} k x^2\), where \(k\) is the spring constant and \(x\) the compression distance.

This captured energy is what's imparted to the potato, propelling it forward upon release. Delving into the potential energy helps us understand how objects store energy that can later be transformed into kinetic energy for doing work — in this case, launching the potato. It also lays the groundwork to explore how different types of potential energy, like gravitational potential energy, play a role in everyday phenomena.