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Chapter 7: Problem 4

Two objects have equal kinetic energies. Are the magnitudes of their momenta equal? Explain your answer.

Short Answer

Expert verified
No, the magnitudes of their momenta are not necessarily equal. Even though they have equal kinetic energies, the object with less mass would have a higher velocity to compensate, and since momentum is a product of mass and velocity, this could result in a greater momentum for the lighter object.

Step by step solution

01

Understand the formulas

The equation for kinetic energy is \( KE = \frac{1}{2} m v^2 \), hence kinetic energy is dependent on both the mass and the square of the velocity of an object. The equation for momentum is \( p = m v \), which shows that momentum is a direct product of the mass and velocity of the object.
02

Reasoning and Comparison

Suppose two objects have equal kinetic energies. From the formula for kinetic energy, a lighter object (smaller mass 'm') would need to have a higher velocity (v) to equal the kinetic energy of a heavier object. Despite having equal kinetic energies, this results in the lighter object having a greater momentum than the heavier object as \( p=mv \), where velocity (v) component is more than mass (m). Hence, the momenta of the two bodies will not be the same.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kinetic Energy
Kinetic energy is the energy of motion. Every moving object has kinetic energy depending on its mass and velocity. The formula for kinetic energy is given by \( KE = \frac{1}{2} mv^2 \).This equation highlights two important factors:
  • Mass (m): Heavier objects (larger mass) will have more kinetic energy if all else is equal.
  • Velocity (v): The velocity is squared, meaning changes in velocity have a significant impact on kinetic energy.
Breaking it down, doubling the velocity actually quadruples the kinetic energy due to the square relationship. Thus, small changes in speed can result in large changes in kinetic energy.
In our exercise, two objects have the same kinetic energy, meaning adjustments occur among their mass and velocity to balance this equation.
Momentum
Momentum is a concept that describes how much motion an object has. This is useful for understanding collisions and movements. The formula is straightforward: \( p = mv \). This tells us that momentum is the product of an object鈥檚 mass and velocity.
There are a few key points to keep in mind about momentum:
  • Mass: More mass means more momentum, assuming the same velocity.
  • Velocity: Faster objects (higher velocity) also translate to higher momentum, assuming the same mass.
  • Momentum is a vector quantity, having both magnitude and direction.
In our exercise, even though the two objects have the same kinetic energy, their momenta vary. This is because the mass and velocity interplay differently for each object. One may be lighter but moves faster, affecting the overall momentum.
Mass and Velocity Relationship
The relationship between mass and velocity is pivotal for understanding kinetic energy and momentum attributes. Both kinetic energy and momentum rely on these variables, but in differing ways.
  • Inverse Relationship for Kinetic Energy: A lighter object must have a greater velocity to achieve the same kinetic energy as a heavier object. This is because kinetic energy relies more heavily on velocity due to the squared factor.
  • Direct Relationship for Momentum: Conversely, the product of mass and velocity directly affects momentum. If you increase either mass or velocity, the momentum goes up evenly.
When two objects have the same kinetic energy, it is rarely the case that they also share the same momentum because the velocity needed to balance the kinetic energy equation skews the momentum values differently. One object might have smaller mass but a much greater velocity to reach the same energy level, thus having different momentum from another object with more mass and less velocity.

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Most popular questions from this chapter

A \(0.10-\mathrm{kg}\) firecracker is hanging by a light string from a tree limb. The fuse is lit and the firecracker explodes into three pieces: a small piece \((0.01 \mathrm{~kg})\), a medium-sized piece \((0.03 \mathrm{~kg})\), and a large piece \((0.06 \mathrm{~kg})\). Assume the firecracker is at the origin of a coordinate system such that the \(+z\) axis points straight up, the \(+x\) axis points due east, and the \(+y\) axis points due north. The fragments fly off according to the following momentum vectors: Largest fragment: \(\quad \vec{p}_{L}=p_{\mathrm{Lx}} \hat{x}-p_{\mathrm{L}} \hat{z}\) Medium fragment: \(\bar{p}_{\mathrm{M}}=p_{\mathrm{M} y} \hat{y}+p_{\mathrm{M} z} \hat{z}\) Smallest fragment: \(\bar{p}_{5}=-p_{5 x} \hat{x}-p_{5 y} \hat{y}\) After the explosion, the component of velocity of the smallest piece in both the \(x\) and \(y\) directions is \(4 \mathrm{~m} / \mathrm{s}\). \(\left(v_{\mathrm{S} x}=v_{\mathrm{S} y}=4 \mathrm{~m} / \mathrm{s}\right.\). ) The velocity of the largest fragment is \(1 \mathrm{~m} / \mathrm{s}\) in the \(z\) direction \(\left(v_{\mathrm{Lz}}=1 \mathrm{~m} / \mathrm{s}\right)\). Determine the velocity vector of each fragment immediately after the explosion.

One ball has four times the mass and twice the speed of another. (a) How does the momentum of the more massive ball compare to the momentum of the less massive one? (b) How does the kinetic energy of the more massive ball compare to the kinetic energy of the less massive one?

Sports A baseball bat strikes a ball when both are moving at \(31.3 \mathrm{~m} / \mathrm{s}\) (relative to the ground) toward each other. The bat and ball are in contact for \(1.20 \mathrm{~ms}\), after which the ball is traveling at a speed of \(42.5 \mathrm{~m} / \mathrm{s}\). The mass of the bat and the ball are \(850 \mathrm{~g}\) and \(145 \mathrm{~g}\), respectively. Find the magnitude and direction of the impulse given to (a) the ball by the bat and (b) the bat by the ball. (c) What average force does the bat exert on the ball? (d) Why doesn't the force shatter the bat?

A \(0.170-\mathrm{kg}\) ball is moving at \(4.00 \mathrm{~m} / \mathrm{s}\) toward the right. It collides elastically with a \(0.155-\mathrm{kg}\) ball moving at \(2.00 \mathrm{~m} / \mathrm{s}\) toward the left. Determine the final velocities of the balls after the collision.

A 2-kg object is moving east at \(4 \mathrm{~m} / \mathrm{s}\) when it collides with a \(6-\mathrm{kg}\) object that is initially at rest. After the completely elastic collision, the larger object moves east at \(1 \mathrm{~m} / \mathrm{s}\). Find the final velocity of the smaller object after the collision.
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