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Chapter 7: Problem 86

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!)

Short Answer

Expert verified
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 by step solution

01

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
02

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
03

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
04

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.
05

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.

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