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, \(m_{\mathrm{p}}\), is \(0.850 \mathrm{~kg} .\) The cannon's spring (with spring constant \(\left.k_{\mathrm{c}}=7.06 \cdot 10^{3} \mathrm{~N} / \mathrm{m}\right)\) 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 \(v_{\mathrm{p}}=175 \mathrm{~m} / \mathrm{s}\). 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, \(\mathrm{v}_{c}\), 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 magnitude and direction of the cannon's velocity after firing the potato are 14.88 m/s to the left. The total mechanical energy before firing is 28.24*10^3 J, and the total mechanical energy after firing is 14.09*10^3 J.

Step-by-step solution

Determine initial and final linear momentum

Initially, the system is at rest, so the total linear momentum is zero. After the launch, the cannon moves in the opposite direction of the potato. Let's assume that positive direction is the direction of the potato's velocity. Using the conservation of linear momentum, we can write:$$ m_{p}v_{p} + m_{c}v_{c} = 0 $$Now we can solve for the cannon's velocity, \(v_{c}\).

Calculate cannon's velocity

Use the formula from Step 1 and solve for \(v_c\):$$ v_{c} = -\frac{m_{p}v_{p}}{m_{c}} $$Plug in the given values for the mass of the potato (\(m_p = 0.850 \, kg\)), velocity of the potato (\(v_p = 175 \, m/s\)), and mass of the cannon (\(m_c = 10.0 \, kg\)):$$ v_c = -\frac{(0.850 \, kg)(175 \, m/s)}{10.0 \, kg} = -14.88 \, m/s $$The negative sign indicates that the cannon's velocity is opposite to the direction of the potato's velocity. Therefore, the magnitude and direction of the cannon's velocity are \(14.88 \, m/s\) to the left.

Calculate initial and final mechanical energies

Initially, the only energy in the system is the potential energy stored in the compressed spring (\(U_{c} = \frac{1}{2}k_{c}x^{2}\)), where \(x\) is the compression length. After firing, the system has kinetic energies for both the potato (\(K_{p}\)) and the cannon (\(K_{c}\)). The total mechanical energy for each case will be given by:$$ E_{initial} = U_{c} $$and$$ E_{final} = K_{p} + K_{c} $$We can calculate \(U_{c}\) using the spring constant, \(k_{c} = 7.06*10^3 \, N/m\), and compression length, \(x = 2.00 \, m\):$$ U_{c} = \frac{1}{2}(7.06 * 10^3 \, N/m)(2.00 \, m)^2 = 28.24*10^3 \, J $$Now we can calculate \(K_{p}\) and \(K_{c}\) using the mass and velocity of the potato and the cannon respectively:$$ K_{p} = \frac{1}{2} m_{p} v_{p}^2 = \frac{1}{2}(0.850\, kg)(175 \,m/s)^2 = 12.98*10^3 \, J $$$$ K_{c} = \frac{1}{2} m_{c} v_{c}^2 = \frac{1}{2}(10.0\, kg)(14.88 \,m/s)^2 = 1.11 * 10^3 \, J $$Adding \(K_{p}\) and \(K_{c}\) gives the final total mechanical energy:$$ E_{final} = K_{p} + K_{c} = 12.98*10^3 \, J + 1.11*10^3 \, J = 14.09*10^3 \, J $$Now we have the total mechanical energies before and after firing the potato:$$ E_{initial} = 28.24*10^{3}\, J $$$$ E_{final} = 14.09*10^{3}\, J $$a) Velocity of the cannon: \(v_c = -14.88 \, m/s\) (leftward) b) Total mechanical energy before firing: \(E_{initial} = 28.24*10^{3}\, J\), Total mechanical energy after firing: \(E_{final} = 14.09*10^{3}\, J\)

Key concepts

Conservation of Linear Momentum

The principle of conservation of linear momentum is a fundamental law of physics stating that if no external force acts on a system, the total linear momentum of the system remains constant in time. It's crucial in analyzing collision and explosion problems, like the launch of a potato from a cannon. Physicists use this principle to predict the resulting motion of objects without knowing the details of the forces between them.

As demonstrated in the solution, if the system of the potato cannon is initially at rest, its initial momentum is zero. After the potato is fired, both the cannon and the potato move in opposite directions, but their combined momentum still equals zero. This demonstrates momentum conservation—the momentum lost by the potato being shot in one direction is gained by the cannon moving in the other. Therefore, the velocity (\(v_c\)) of the cannon (after the potato is shot) can be determined by rearranging the equation to solve for the cannon's velocity.

Mechanical Energy

Mechanical energy is the sum of kinetic and potential energy in an object or system. When the forces involved are conservative, mechanical energy is conserved; however, if non-conservative forces (like friction) are present, mechanical energy can change. In the context of the potato cannon exercise, ignoring friction allows us to assume that the mechanical energy is entirely converted from the potential energy stored in the spring to the kinetic energy of the potato and cannon.

The solution calculates this conversion starting with the potential energy in the spring and finishing with the kinetic energy of the potato and cannon post-launch. By comparing initial and final mechanical energies, one can determine if mechanical energy is conserved or if other forces have played a role.

Kinetic Energy

Kinetic energy is the energy an object possesses because of its motion. It's given by the equation \( K = \frac{1}{2}mv^2 \), where \( m \) is the mass of the object and \( v \) is its velocity. As the object's velocity increases, the kinetic energy increases quadratically. In the potato cannon situation, when the spring releases its stored energy, it accelerates both the potato and the cannon, giving them kinetic energy. The solution illustrates how to calculate this energy for both objects after launch, and emphasize the dependence of kinetic energy on both mass and velocity.

Potential Energy

Potential energy is the energy stored within a system because of the position or configuration of different parts. For the potato cannon, the compressed spring possesses elastic potential energy, which is given by the equation \( U = \frac{1}{2}kx^2 \), where \( k \) is the spring constant and \( x \) is the compression distance. This type of potential energy is significant because it can be converted into kinetic energy when the spring is released. The amount of energy stored in the spring, and thus the potential energy of the system, directly relates to how much the spring is compressed.

Spring Constant

The spring constant, denoted as \( k \), is a measure of a spring's stiffness. It tells us how much force is needed to compress or stretch the spring by a certain distance. The stiffer the spring, the higher its spring constant. In the context of the exercise, a high spring constant indicates that the spring is stiff and can store significant energy when compressed, which in turn means that the potato will be launched with a considerable velocity. The potential energy in the spring, calculated using the spring constant, is what propels the potato with the calculated kinetic energy and also gives the cannon its kinetic energy in the opposite direction.