106_PartUniversity Physics Solution

106_PartUniversity Physics Solution - 3-36 Chapter 3 9.80...

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3-36 Chapter 3 (d) Relative to the flat car, the ball is projected at an angle 1 9.80 m/s tan 65 . 4.54 m/s θ = = ° Relative to the ground the angle is 1 9.80 m/s tan 35.7 4.54 m/s 9.10 m/s θ = = ° + . E VALUATE : In both frames of reference the ball moves in a parabolic path with 0 x a = and y a g = − . The only difference between the description of the motion in the two frames is the horizontal component of the ball°s velocity. 3.87. I DENTIFY : The pellets move in projectile motion. The vertical motion determines their time in the air. S ET U P : 0 0 cos1.0 x v v = ° , 0 0 sin1.0 y v v = ° . E XECUTE : (a) 0 2 y v t g = . 0 0 x x x v t = gives 0 0 0 sin1.0 ( cos1.0 80 m v x x v g = = 2 ° °) . (b) The probability is 1000 times the ratio of the area of the top of the person°s head to the area of the circle in which the pellets land. 2 2 3 2 (10 10 m) (1000) 1.6 10 . (80 m) π π × = × (c) The slower rise will tend to reduce the time in the air and hence reduce the radius. The slower horizontal velocity will also reduce the radius. The lower speed would tend to increase the time of descent, hence increasing the radius. As the bullets fall, the friction effect is smaller than when they were rising, and the overall effect is to decrease the radius. E VALUATE : The small angle of deviation from the vertical still causes the pellets to spread over a large area because their time in the air is large. 3.88. I DENTIFY : Write an expression for the square of the distance 2 ( ) D from the origin to the particle, expressed as a function of time. Then take the derivative of 2 D with respect to t , and solve for the value of t when this derivative is zero. If the discriminant is zero or negative, the distance D will never decrease. S ET U P : 2 2 2 D x y = + , with ( ) x t and ( ) y t given by Eqs.(3.20) and (3.21). E XECUTE : Following this process, 1 sin 8/9 70.5 . = ° E VALUATE : We know that if the object is thrown straight up it moves away from P and then returns, so we are not surprised that the projectile angle must be less than some maximum value for the distance to always increase with time. 3.89. I DENTIFY : The baseball moves in projectile motion. S ET U P : Use coordinates where the x -axis is horizontal and the y -axis is vertical. E XECUTE : (a) The trajectory of the projectile is given by Eq. (3.27), with 0 θ φ , α = + and the equation describing the incline is tan . y x θ = Setting these equal and factoring out the 0 x = root (where the projectile is on the incline) gives a value for 0 ; x the range measured along the incline is 2 2 0 2 cos ( ) /cos [tan( ) tan ] . cos v θ φ x θ θ φ g θ θ + = + (b) Of the many ways to approach this problem, a convenient way is to use the same sort of substitution, involving double angles, as was used to derive the expression for the range along a horizontal incline. Specifically, write the above in terms of θ φ , α = + as 2 2 0 2 2 [sin cos cos cos sin ] .
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