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Chapter 5: Problem 5

A \(0.53-\mathrm{kg}\) billiard ball initially at rest is given a speed of \(12 \mathrm{m} / \mathrm{s}\) during a time interval of \(4.0 \mathrm{ms}\). What average force acted on the ball during this time?

Short Answer

Expert verified
The average force is 1590 N.

Step by step solution

01

Identify the Given Values

We are given a billiard ball of mass \(m = 0.53\, \mathrm{kg}\), an initial speed \(u = 0\, \mathrm{m/s}\), a final speed \(v = 12 \mathrm{m/s}\), and a time interval \(\Delta t = 4.0 \mathrm{ms} = 4.0 \times 10^{-3} \mathrm{s}\). We need to find the average force \(F_{\text{avg}}\) acting on the ball.
02

Use Newton's Second Law

According to Newton's Second Law, the force is given by the change in momentum over time: \(F_{\text{avg}} = \frac{\Delta p}{\Delta t}\). Here, \(\Delta p\) is the change in momentum, \(\Delta p = m(v-u)\).
03

Calculate Change in Momentum

Substitute the given values into the momentum change equation: \(\Delta p = 0.53\, \mathrm{kg} \times (12 \mathrm{m/s} - 0 \mathrm{m/s})\)\(\Delta p = 0.53 \times 12 = 6.36 \mathrm{kg\cdot m/s}\).
04

Calculate Average Force

Plug the change in momentum and time interval into the force formula: \[F_{\text{avg}} = \frac{6.36\, \mathrm{kg \cdot m/s}}{4.0 \times 10^{-3} \, \mathrm{s}}\]\[F_{\text{avg}} = 1590 \, \mathrm{N}\].

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Key Concepts

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

Momentum
Momentum is a key concept in physics that helps us understand how objects move and interact with each other. It refers to the quantity of motion an object has and is given by the product of an object's mass and velocity. Momentum is typically represented by the symbol \( p \) and is calculated using the formula:
  • \( p = m \cdot v \)
where \( m \) is the mass of the object and \( v \) is its velocity. The SI unit of momentum is kg路m/s.
In the case of our billiard ball, it starts from rest, meaning its initial momentum is zero. When it is struck and reaches a velocity of \( 12 \text{ m/s} \), its new momentum is \( 6.36 \text{ kg路m/s} \). This change in momentum is crucial, as it is directly linked to the force applied during the collision.
Average Force
Average force is an important concept derived from Newton's Second Law. It represents the constant force that would produce the same effect on an object's motion as a varying force averaged over a time interval. This can be calculated by finding the change in momentum divided by the time taken for this change. Mathematically, average force is expressed as:
  • \( F_{\text{avg}} = \frac{\Delta p}{\Delta t} \)
Here's how it works in our scenario. The momentum of the billiard ball changes from \( 0 \) to \( 6.36 \text{ kg路m/s} \) over a brief time of \( 0.004 \text{ s} \). Plugging these values into the formula, the average force is calculated to be approximately \( 1590 \text{ N} \).
This value indicates the strong and rapid force applied to the ball to achieve the new momentum.
Time Interval
The time interval is a critical component when calculating changes in motion. It is the duration over which an event, such as a force being applied, occurs. In our example, the billiard ball is subject to an external force for only a very short period of \( 4.0 \text{ ms} \) (milliseconds), which equals \( 0.004 \text{ seconds} \).
This brief time is essential in determining how large the average force must be to change the ball's motion significantly in such a short span. The shorter the time interval for a given change in momentum, the greater the average force must be. Understanding time intervals allows us to predict and explain the effects of forces in dynamic situations accurately.

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Most popular questions from this chapter

The Fall of \(T\). rex Paleontologists estimate that if a Tynamosaurus rex were to trip and fall, it would have experienced a force of approximately \(260,000 \mathrm{N}\) acting on its torso when it hit the ground. Assuming the torso has a mass of \(3800 \mathrm{kg},\) (a) find the magnitude of the torso's upward acceleration as it comes to rest. (For comparison, humans lose consciousness with an acceleration of about \(7 g .\) (b) Assuming the torso is in free fall for a distance of \(1.46 \mathrm{m}\) as it falls to the ground, how much time is required for the torso to come to rest once it contacts the ground?

At the local grocery store, you push a 14.5 -kg shopping cart. You stop for a moment to add a bag of dog food to your cart. With a force of \(12.0 \mathrm{N}\), you now accelerate the cart from rest through a distance of \(2.29 \mathrm{m}\) in \(3.00 \mathrm{s}\). What was the mass of the dog food?

\- ce Predict/Explain You drop two balls of equal diameter from the same height at the same time. Ball 1 is made of metal and has a greater mass than ball \(2,\) which is made of wood. The upward force due to air resistance is the same for both balls. (a) Is the drop time of ball 1 greater than, less than, or equal to the drop time of ball \(2 ?\) (b) Choose the best explanation from among the following: I. The acceleration of gravity is the same for all objects, regardless of mass. II. The more massive ball is harder to accelerate. III. Air resistance has less effect on the more massive ball.

\- - IP \(A\) 42.0-kg parachutist is moving straight downward with a speed of \(3.85 \mathrm{m} / \mathrm{s}\). (a) If the parachutist comes to rest with constant acceleration over a distance of \(0.750 \mathrm{m},\) what force does the ground exert on her? (b) If the parachutist comes to rest over a shorter distance, is the force exerted by the ground greater than, less than, or the same as in part (a)? Explain.

You hold a brick at rest in your hand. (a) How many forces act on the brick? (b) Identify these forces, (c) Are these forces equal in magnitude and opposite in direction? (d) Are these forces an action-reaction pair? Explain.
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