Ch 07: Newton's Third Law

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 07: Newton's Third Law

Problem 32b

Chapter 7, Problem 32b

A 75 kg archer on ice skates is standing at rest on very smooth ice. He shoots a 450 g arrow horizontally. When released, the arrow reaches a speed of 110 m/s in 0.25 s. Assume that the force of the bow string on the arrow is constant. What is the archer's recoil speed?

Verified step by step guidance

Step 1: Identify the principle of conservation of momentum. Since the archer and the arrow are initially at rest, their total momentum before the shot is zero. After the arrow is shot, the momentum of the archer and the arrow must still sum to zero because there are no external forces acting horizontally.

Step 2: Calculate the momentum of the arrow after it is shot. Momentum is given by the formula:

p = m v

, where

m

is the mass of the arrow (450 g or 0.450 kg) and

v

is its velocity (110 m/s).

Step 3: Apply the conservation of momentum. The momentum of the archer must be equal in magnitude but opposite in direction to the momentum of the arrow. Let the recoil speed of the archer be

v

_r. The equation becomes:

m_{a} v_{a} + m_{r} v_{r} = 0

, where

m_{a}

and

v_{a}

are the mass and velocity of the arrow, and

m_{r}

and

v_{r}

are the mass and recoil speed of the archer.

Step 4: Rearrange the equation to solve for the recoil speed of the archer:

v_{r} = -(m_{a} v_{a})/ m_{r}

. Substitute the values:

m_{a} = 0.450

kg,

v_{a} = 110

m/s, and

m_{r} = 75

kg.

Step 5: Perform the substitution and simplify the expression to find the recoil speed of the archer. Remember that the negative sign indicates the direction of the recoil is opposite to the direction of the arrow's motion.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Conservation of Momentum

The principle of conservation of momentum states that in a closed system, the total momentum before an event must equal the total momentum after the event. In this scenario, the archer and the arrow form a closed system. When the arrow is shot, the momentum gained by the arrow must be equal and opposite to the momentum gained by the archer, leading to the calculation of the archer's recoil speed.

Momentum

Momentum is defined as the product of an object's mass and its velocity, represented mathematically as p = mv. In this case, both the archer and the arrow have momentum, and their respective masses and velocities will determine how they move after the arrow is released. Understanding how to calculate momentum is crucial for solving the problem of the archer's recoil speed.

Newton's Third Law of Motion

Newton's Third Law states that for every action, there is an equal and opposite reaction. When the archer shoots the arrow, the force exerted on the arrow results in an equal force acting on the archer in the opposite direction. This law is fundamental in understanding the interaction between the archer and the arrow, allowing us to determine the archer's recoil speed based on the force applied to the arrow.

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