Force on a Baseball

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Impulse and Momentum

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Impulse Newton s second law of motion, F ma, can be rewritten by using the definition of acceleration as the change in velocity divided by the time needed to make that change It can be represented by the following equation: F ma m

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Multiplying both sides of the equation by the time interval, t, results in the following equation: F t m v Impulse, or F t, is the product of the average force on an object and the time interval over which it acts Impulse is measured in newton-seconds For instances in which the force varies with time, the magnitude of an impulse is found by determining the area under the curve of a force-time graph, such as the one shown in Figure 9-1 The right side of the equation, m v, involves the change in velocity: v vf vi Therefore, m v mvf mvi The product of the object s mass, m, and the object s velocity, v, is defined as the momentum of the object Momentum is measured in kg m/s An object s momentum, also known as linear momentum, is represented by the following equation Momentum p mv

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The momentum of an object is equal to the mass of the object times the object s velocity

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Momentum and impulse vectors are orange Force vectors are blue Acceleration vectors are violet Velocity vectors are red Displacement vectors are green

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Recall the equation F t m v mvf mvi Because mvf pf and mvi pi, this equation can be rewritten as follows: F t m v pf pi The right side of this equation, pf pi, describes the change in momentum of an object Thus, the impulse on an object is equal to the change in its momentum, which is called the impulse-momentum theorem The impulse-momentum theorem is represented by the following equation Impulse-Momentum Theorem F t pf pi

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The impulse on an object is equal to the object s final momentum minus the object s initial momentum

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9 Momentum and Its Conservation

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If the force on an object is constant, the impulse is the product of the force multiplied by the time interval over which it acts Generally, the force is not constant, however, and the impulse is found by using an average force multiplied by the time interval over which it acts, or by finding the area under a force-time graph Because velocity is a vector, momentum also is a vector Similarly, impulse is a vector because force is a vector This means that signs will be important for motion in one dimension

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Using the Impulse-Momentum Theorem

What is the change in momentum of a baseball From the impulsemomentum theorem, you know that the change in momentum is equal to the impulse acting on it The impulse on a baseball can be calculated by using a force-time graph In Figure 9-1, the area under the curve is approximately 131 N s The direction of the impulse is in the direction of the force Therefore, the change in momentum of the ball also is 131 N s Because 1 N s is equal to 1 kg m/s, the momentum gained by the ball is 131 kg m/s in the direction of the force acting on it Assume that a batter hits a fastball Before the collision of the ball and bat, the ball, with a mass of 0145 kg, has a velocity of 38 m/s Assume that the positive direction is toward the pitcher Therefore, the baseball s momentum is pi (0145 kg)( 38 m/s) 55 kg m/s What is the momentum of the ball after the collision Solve the impulse-momentum theorem for the final momentum: pf pi F t The ball s final momentum is the sum of the initial momentum and the impulse Thus, the ball s final momentum is calculated as follows pf pi 131 kg m/s 55 kg m/s 131 kg m/s 76 kg m/s mvf, solving for

Running Shoes Running is hard on the feet When a runner s foot strikes the ground, the force exerted by the ground on it is as much as four times the runner s weight The cushioning in an athletic shoe is designed to reduce this force by lengthening the time interval over which the force is exerted