Problem 86 - University Physics with Modern Physics

Question

A golf ball of mass \(45.0 \mathrm{~g}\) moving at a speed of \(120 . \mathrm{km} / \mathrm{h}\) collides head on with a French TGV high-speed train of mass \(3.8 \cdot 10^{5} \mathrm{~kg}\) that is traveling at \(300 . \mathrm{km} / \mathrm{h}\). Assuming that the collision is elastic, what is the speed of the golf ball after the collision? (Do not try to conduct this experiment!)

Step-by-Step Solution

Answer

Answer: To find the final velocity of the golf ball after the collision, follow the step-by-step solution provided, which involves converting units to SI units, using conservation of momentum and kinetic energy, solving for v1', and converting the final velocity back to km/h if needed. By solving the equations with the given values, you will find the final velocity of the golf ball after the collision.

Step 1: Convert units to SI units

Before doing any calculations, convert the given mass and velocities into SI units. The mass of golf ball is given in grams (g), so convert it into kilograms (kg). The velocities are given in kilometers per hour (km/h), so convert them into meters per second (m/s). Conversion factors: 1 kg = 1000 g 1 km/h = 1000 m/3600 s Mass of golf ball (m1) = 45.0 g = 45.0/1000 kg = 0.045 kg Velocity of golf ball (v1) = 120 km/h = 120 * (1000/3600) m/s = 33.3 m/s Mass of train (m2) = 3.8 * 10^5 kg (already in SI units) Velocity of train (v2) = 300 km/h = 300 * (1000/3600) m/s = 83.3 m/s

Step 2: Use conservation of momentum

For an elastic collision, the momentum before and after the collision is conserved. The total momentum before collision (P_initial) is the sum of the momentum of golf ball and the train's momentum. The total momentum after the collision (P_final) can be expressed in terms of their final velocities (v1' and v2'). P_initial = P_final m1*v1 + m2*v2 = m1*v1' + m2*v2' Since we're interested in finding the final velocity of the golf ball (v1'), we can rearrange the equation to isolate v1' on one side: v1' = (m1*v1 + m2*v2 - m2*v2')/m1

Step 3: Use conservation of kinetic energy

For an elastic collision, the total kinetic energy before the collision is equal to the total kinetic energy after the collision. We can express the initial and final kinetic energy in terms of the initial and final velocities: KE_initial = KE_final 0.5*m1*v1^2 + 0.5*m2*v2^2 = 0.5*m1*(v1')^2 + 0.5*m2*(v2')^2 Our goal is to find v1', so we need to eliminate v2' from this equation. To do that, let's first solve for v2' from the momentum equation: v2' = (m1*v1 + m2*v2 - m1*v1')/m2 Now, substitute this expression for v2' into the kinetic energy equation. This will give us an equation only in terms of v1': 0.5*m1*v1^2 + 0.5*m2*v2^2 = 0.5*m1*(v1')^2 + 0.5*m2*((m1*v1 + m2*v2 - m1*v1')/m2)^2

Step 4: Solve for the final velocity of the golf ball (v1')

Now, we have a single equation containing only v1'. Solve this equation for v1'. This might take a few steps involving algebraic manipulation, expanding, and simplifying. Note that dividing the whole equation by 0.5 will make it easier to deal with. Then expand and simplify the equation until you isolate v1' on one side of the equation. Once you have an expression for v1' in terms of known values, plug in the values for m1, m2, v1, and v2, and solve for v1'. This will give you the final velocity of the golf ball after the collision.

Step 5: Convert the final velocity back to km/h (optional)

Once you have found the final velocity of the golf ball in meters per second (m/s), you can convert the velocity back to kilometers per hour (km/h) by using the conversion factor: v1' (km/h) = v1' (m/s) * (3600/1000) This gives you the final velocity of the golf ball in the original units.

Key Concepts

Conservation of Momentum

In physics, the principle of conservation of momentum plays a crucial role in analyzing collisions, such as the case of the golf ball and the train. Momentum is defined as the product of an object's mass and velocity. It's a vector quantity, having both magnitude and direction.

The conservation of momentum states that the total momentum of a closed system remains constant if no external forces act upon it. For an elastic collision, which is a type of collision where both kinetic energy and momentum are conserved, this principle is particularly important.

• The total momentum before a collision is equal to the total momentum after the collision.
• In mathematical terms, this can be written as: \[ m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2' \] where \( m_1 \) and \( m_2 \) are the masses of the colliding objects, and \( v_1, v_2, v_1', \) and \( v_2' \) represent their initial and final velocities.

Knowing the initial velocities and masses allows us to calculate the final velocities of the objects after the collision. This is especially useful in solving problems involving collisions, to determine the aftermath of the interaction.

Conservation of Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion, and it's calculated using the formula \( KE = \frac{1}{2}mv^2 \). In an elastic collision, such as the one between the golf ball and the train, the total kinetic energy is conserved. This means it remains the same before and after the collision.

For conservation of kinetic energy in a collision, the sum of the kinetic energies of all bodies involved, before the collision, equals the sum after the collision. The equation for this is:
\[ 0.5m_1v_1^2 + 0.5m_2v_2^2 = 0.5m_1v_1'^2 + 0.5m_2v_2'^2 \]

• In this expression, \( m_1, v_1, v_1' \) similarly refer to the mass and initial, final velocities of one object, while \( m_2, v_2, v_2' \) refer to those of the second object.
• The conservation of kinetic energy helps determine the aftermath of an elastic collision by allowing one to solve the equations to calculate unknown variables, such as final velocities.

Understanding these concepts is key to solving problems related to elastic collisions effectively. They provide insight into how energy and motion are distributed and conserved during such interactions.

SI Unit Conversion

When dealing with physics problems, especially those involving calculations like in our collision scenario, it's essential to use a consistent set of units, typically the International System of Units (SI). This ensures accuracy and clarity in your results.
SI units are the standard for scientific calculations: mass is measured in kilograms (kg), velocity in meters per second (m/s), and so on.

• For example, converting mass from grams to kilograms requires dividing by 1000: \( 45.0 \text{ g} = 0.045 \text{ kg} \).
• Velocity, given initially in kilometers per hour, must be converted to meters per second using the conversion factor: \[ 1 \text{ km/h} = \frac{1000 \text{ m}}{3600 \text{ s}} \] This equates to multiplying the velocity by \( \frac{1000}{3600} \).

Adopting SI units from the beginning assists in applying physical laws correctly, enabling straightforward manipulation of equations to find the solution. Without it, calculations might be prone to errors due to inconsistent units, complicating the process and results.