The Physics Behind Collisions: Impulse & Momentum

What really happens when you hit a ball with a bat? This question can unravel the fascinating physics of collisions. When the bat strikes the ball, the interaction lasts just a microsecond, but within this brief period, significant physical changes occur. Let’s delve into the details of these changes and understand the underlying principles of force and momentum.

Force and Momentum During Collision

When the bat contacts the ball, a force F̅ is exerted on the ball, causing it to decelerate, stop, and potentially reverse direction. The force is not constant; it starts at zero before contact, peaks when the bat and ball are fully in contact and diminishes as they separate. This varying force changes the ball's linear momentum p̅.

According to Newton’s second law, the relationship between force and momentum is expressed as:

F̅ = dp̅/dt

Rearranging this equation gives us the momentum change over a small time interval dt:

dp̅ = F̅(t) dt

To find the total change in momentum during the collision from initial time tᵢ to final time t_f, we integrate both sides:

∫ dp̅ from tᵢ to t_f = ∫ F̅(t) dt from tᵢ to t_f

The left side represents the change in momentum, Δp̅ = p̅_f - p̅ᵢ, while the right side represents the impulse J̅ of the collision:

J̅ = ∫ F̅ dt from tᵢ to t_f

Thus, the change in an object’s momentum equals the impulse it experiences. This principle is known as the linear momentum–impulse theorem:

Δp̅ = J̅

This theorem indicates that impulse is a vector quantity and its direction matches the change in momentum's direction.

Calculating Impulse from Force-Time Graphs

Impulse can be visualized using a force versus time graph. The area under the curve represents the impulse J̅. If the force varies, we may use the average force Fₐᵥ and the time interval Δt to find the impulse:

J = Fₐᵥ Δt

Even if the force changes over time, the total impulse can be calculated by finding the area under the force-time curve.

Newton’s Third Law and Opposite Impulses

Newton’s third law tells us that the force exerted on the bat by the ball is equal in magnitude but opposite in direction to the force exerted on the ball by the bat. Therefore, the impulse on the bat is equal in magnitude but opposite in direction to the impulse on the ball.

Practical Example: Ball Collision with a Wall

Consider a 0.40 kg ball thrown against a brick wall, moving at 30 m/s and rebounding at 20 m/s. We need to determine the impulse on the ball and the average horizontal force exerted by the wall if the collision lasts 0.01 s.

Calculating Initial and Final Momentum

Initial momentum:

p₁ = mv₁ = (0.40 kg)(−30 m/s) = −12 kg·m/s

Final momentum:

p₂ = mv₂ = (0.40 kg)(+20 m/s) = +8.0 kg·m/s

Determining Impulse

Impulse is the change in momentum:

J_ball = p₂ - p₁ = 8.0 kg·m/s − (−12 kg·m/s) = 20 kg·m/s

Finding Average Force

Using the impulse-momentum theorem:

F_avg = J/Δt = 20 N·s / 0.010 s = 2000 N

The positive sign of the impulse indicates the direction is to the right, as expected. This "kick" from the wall propels the ball to the right.

Comparing Momentum and Kinetic Energy

Understanding the difference between momentum and kinetic energy can be challenging. The impulse-momentum theorem states that a change in a particle’s momentum is due to impulse, dependent on the time over which the force acts. The work-energy theorem states that kinetic energy changes when work is done, dependent on the distance over which the force acts, not the time.

Example Comparison

If given the choice between catching a 0.10-kg ball moving at 20 m/s or a 0.50-kg ball moving at 4.0 m/s, which would be easier? Both have the same momentum (p = 2.0 kg·m/s), but different kinetic energies. The 0.50-kg ball has a kinetic energy of 4.0 J, while the 0.10-kg ball has 20 J.

Both require the same impulse to stop, but stopping the 0.10-kg ball requires five times more work because it has five times more kinetic energy. Using the same force to stop both, the time taken would be the same, but the hand will be pushed back more by the faster-moving smaller ball. Thus, catching the 0.50-kg ball, which has lower kinetic energy, is easier.

Conclusion

Understanding the physics of bat-ball collisions involves concepts of impulse and momentum, along with Newton's laws. By analyzing force-time graphs and applying these principles, we gain a clearer understanding of the physical interactions during collisions. This knowledge is essential for solving physics problems and appreciating the dynamics of forces and motions in everyday life.