Fall 07 Midterm 1 Solutions Ch. 1-3

Fall 07 Midterm 1 Solutions Ch. 1-3 - Note For all problems...

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Unformatted text preview: Note: For all problems on this exam, ignore air resistance and use g = 9.80 m/s2 for the acceleration of gravity when necessary. 1. Dimensional Analysis and Vectors [8 pts] a. Newton's Law of Gravitation says that the force F on a mass m1 due to the gravitational influence of another mass m2 at a distance r is F = Gm1m2/r2, where G is a universal constant. Force has dimensions (mass) (length)/(time)2, or in abbreviated form, ML/T2. What are the dimensions of the constant G? [4 pts] ML/T2 = G M M/L2 G = L3/(M T2) L3/(MT2) b. As shown in the diagram, vector A has magnitude 3.00 units and is perpendicular to vector B, with magnitude 4.00 units. Vector C has magnitude 6.00 units. The angle = 30.0. Find the vector A+B+C. [4 pts] y A = 3.00 i B = +4.00 j C = 6.00 (sin i cos j) = +3.00 i 5.20 j A + B + C = 1.20 j B A C x 1.20 j A 2. Velocity and Speed [8 pts] a. Two campers plan to start a hike at a trailhead 3.00 km north of their camp. The first camper wants to pick up a map at the visitor center 5.00 km north of camp before starting the hike, and suggests that they meet at the trailhead. If both walk at a constant speed of 2.00 km/h, how long after the first camper leaves should the second camper wait before starting out, so they reach the trailhead at the same time? [4 pts] Trailhead Campsite 2.00 h b. A robot's position is described by: x(t) = 3.00 m (2.00 m/s) t y(t) = 4.00 m + (1.00 m/s2) t2 Find the robot's average velocity vector between t1 = 2.00 s and t2 = 5.00 s. [4 pts] x = x(5.00 s) x(2.00 s) = (7.00 m) (1.00 m) = 6.00 m y = y(5.00 s) y(2.00 s) = (21.0 m) (0.00 m) = 21.0 m vxavg = x/t = (6.00 m)/(3.00 s) = 2.00 m/s vyavg = y/t = (21.0 m)/(3.00 s) = 7.00 m/s vavg = (2.00 m/s i + 7.00 m/s j) 2.00 m/s i + 7.00 m/s j Path of Camper 2 A Path of Camper 1 Time for first camper to arrive: t1 = d1/v = (7.00 km)/(2.00 km/h) = 3.50 h Travel time for second camper: t2 = d2/v = (3.00 km)/(2.00 km/h) = 1.50 h Time second camper should wait before starting t1 t2 = 2.00 h Visitor Center 3. Acceleration [8 pts] a. An elevator starts from rest on the ground floor, accelerates at 1.00 m/s2 until it reaches its maximum speed of 5.00 m/s, travels for some time at this speed, and then eventually decelerates at 1.00 m/s2 to stop at a floor 110.0 m above. How much time does this elevator ride take? [4 pts] There are three distinct time intervals of motion: constant acceleration, constant velocity, and constant deceleration. The length of the trip is the sum of these three time intervals. Time to reach maximum speed from rest: t1 = (vmax 0)/a = (5.00 m/s)/(1.00 m/s2) = 5.00 s Distance traveled while accelerating: y1 = a (t1)2 = (1.00 m/s2) (5.00 s)2 = 12.5 m The time to go from maximum speed to rest is the same: t3 = (0 vmax)/a = (5.00 m/s)/(1.00 m/s2) = 5.00 s Distance traveled while decelerating is the same too: y3 = a (t3)2 + vmax (t3) = 12.5 m, where a = 1.00 m/s2. The remaining distance is travelled at maximum speed: y2 = y y1 y3 = 110.0 m 12.5 m 12.5 m = 85.0 m This takes a time t2 = y2/vmax = (85.0 m)/(5.00 m/s) = 17.0 s 27.0 s The total time is t = t1 + t2 + t3 = 27.0 s b. An object starts from rest at position x = 0 m at time t = 0 s. Five seconds later, at t = 5.0 s, it is observed to be at x = 40.0 m, with velocity vx = 12 m/s. Was the object's acceleration uniform? Explain your reasoning [3 pts], and sketch a velocity vs. time graph consistent with the data given [1 pt]. If the object were accelerating uniformly, its acceleration would be a = v/t = (12.0 m/s)/(5.0 s) = 2.40 m/s2 Starting from rest with this acceleration, the object would have a displacement of x = a (t)2 = (2.40 m/s2)(5.00 s)2 = 30.0 m Therefore the object is not accelerating uniformly. Under uniform acceleration, the object's velocity graph vx would be a straight line between 0 m/s (at t=0) and 12 m/s (m/s) (at t= 5 s). See the dashed line in the drawing. 12 m/s Since the displacement is larger than expected from uniform acceleration, the velocity graph must be above this line (so the area under it is larger), but pass through the ends, like the solid curve in the drawing. 5 s A t (s) 4. Motion in Two Dimensions [8 pts] a. An archer shoots an arrow horizontally. The arrow lands 60.0 m away from her, and is stuck into the ground at an angle = 3.00 from horizontal. With what speed was the arrow fired? (You do not need to know the height of the archer, or the distance from the ground that the arrow was fired). [4 pts] Arrow The arrow moves with a constant horizontal velocity: vx = v0 cos = v0 Ground The arrow's vertical velocity is initially 0 and changes with time due to the acceleration of gravity: vy(t) = v0 sin g t = g t The angle it strikes the ground is the angle of its final velocity vector: tan = |vy/vx| =(gt)/v0 So it is in flight for a time t = v0 tan/g The horizontal distance traveled is x = vx t = v02 tan/g, so v02 = g x/tan and v0 = [(9.80 m/s2) (60.0 m)/tan(3.00)]1/2 = 106 m/s b. Chad throws a pass to his teammate Mario, who is 15.0 m meters in front of Chad and running directly away from him at 8.00 m/s when the pass is released. Chad throws the pass at a 45.0 angle from the horizontal, and Mario catches it without changing his speed or direction. What was the initial speed of Chad's throw? [4 pts] Let t be the time between the throw and the catch, and x be the initial displacement between Chad and Mario. v0 is the speed of the throw, and vMis Mario's speed. Then the throw travels a horizontal distance: R = x + vM (t) = v0 cos (t) So t = x/(v0 cos vM) In the vertical direction, the ball's net displacement y = 0, so: y = 0 = v0 sin (t) g (t)2 v0sin = g (t) = g (x)/(v0 cos vM) 2v02 sin cos 2v0 vM sin g (x) = 0 = v02 sin2 2 v0 vM sin g(x) Since the throw is at = 45, sin2 = 1, so the quadratic can be simplified to: v02 2 vM sin v0 g(x) = 0 v0 = (2 vM sin [4 vM2 sin2 +4 g(x)]1/2)/2 v0 = (8.00 m/s) sin(45) [(8.00 m/s)2 sin2(45) + (9.80 m/s2)(15.0 m)]1/2 v0 = 19.0 m/s (speed of throw must be the positive solution) 19.0 m/s 106 m/s A 5. Circular and Relative Motion [8 pts] a. The minuteshand of a clock is 5.00 cm long. What is the magnitude of the acceleration at a point on the outer end of the hand? [4 pts] The end of the minutes hand travels a distance 2R in one hour. 1 hour = 60 minutes (60 s/minute) = 3600 s The speed is therefore v = 2R/(3600 s) = 2(0.05 m)/(3600 s) = 8.72665 105 m/s The centripetal acceleration is a = v2/R = (8.72665 105 m/s)2/(0.05 m) = 1.52107 m/s2 1.52107 m/s2 b. A rowboat crosses a 120m wide river that is flowing to the east at a speed of 1.00 m/s. The rower can row at a speed of 2.00 m/s relative to the water. If the rower rows straight north, how far downstream will the boat land on the opposite bank? [4 pts] We call north the positive y direction, and east the positive x direction. The boat takes a time t = y/vy = (120 m)/(2.00 m/s) = 60.0 s to cross the river. In that time, the current carries the boat x = vx (t) = (1.00 m/s) (60.0 s) = 60.0 m downstream. 60.0 m A Note: For all problems on this exam, ignore air resistance and use g = 9.80 m/s2 for the acceleration of gravity when necessary. 1. Dimensional Analysis and Vectors [8 pts] a. Newton's Law of Gravitation says that the force F on a mass m1 due to the gravitational influence of another mass m2 at a distance r is F = Gm1m2/r2, where G is a universal constant. Force has dimensions (mass) (length)/(time)2, or in abbreviated form, ML/T2. What are the dimensions of the constant G? [4 pts] b. As shown in the diagram, vector A has magnitude 3.00 units and is perpendicular to vector B, with magnitude 4.00 units. Vector C has magnitude 6.00 units. The angle = 25.0. Find the vector A+B+C. [4 pts] y B A C x B 2. Velocity and Speed [8 pts] a. Two campers plan to start a hike at a trailhead 3.00 km north of their camp. The first camper wants to pick up a map at the visitor center 5.00 km north of camp before starting the hike, and suggests that they meet at the trailhead. If both walk at a constant speed of 3.75 km/h, how long after the first camper leaves should the second camper wait before starting out, so they reach the trailhead at the same time? [4 pts] Visitor Center Trailhead Campsite b. A robot's position is described by: x(t) = 4.00 m (1.00 m/s) t y(t) = 3.00 m + (2.00 m/s2) t2 Find the robot's average velocity vector between t1 = 2.00 s and t2 = 5.00 s. [4 pts] Path of Camper 2 B Path of Camper 1 3. Acceleration [8 pts] a. An elevator starts from rest on the ground floor, accelerates at 1.00 m/s2 until it reaches its maximum speed of 5.00 m/s, travels for some time at this speed, and then eventually decelerates at 1.00 m/s2 to stop at a floor 140.0 m above. How much time does this elevator ride take? [4 pts] b. An object starts from rest at position x = 0 m at time t = 0 s. Five seconds later, at t = 5.0 s, it is observed to be at x = 45.0 m, with velocity vx = 12 m/s. Was the object's acceleration uniform? Explain your reasoning [3 pts], and sketch a velocity vs. time graph consistent with the data given [1 pt]. vx (m/s) t (s) B 4. Motion in Two Dimensions [8 pts] a. An archer shoots an arrow horizontally. The arrow lands 90.0 m away from her, and is stuck into the ground at an angle = 2.00 from horizontal. With what speed was the arrow fired? (You do not need to know the height of the archer, or the distance from the ground that the arrow was fired). [4 pts] Arrow Ground b. Chad throws a pass to his teammate Mario, who is 15.0 m meters in front of Chad and running directly away from him at 7.50 m/s when the pass is released. Chad throws the pass at a 45.0 angle from the horizontal, and Mario catches it without changing his speed or direction. What was the initial speed of Chad's throw? [4 pts] B 5. Circular and Relative Motion [8 pts] a. The minuteshand of a clock is 6.00 cm long. What is the magnitude of the acceleration at a point on the outer end of the hand? [4 pts] b. A rowboat crosses a 130m wide river that is flowing to the east at a speed of 1.00 m/s. The rower can row at a speed of 2.00 m/s relative to the water. If the rower rows straight north, how far downstream will the boat land on the opposite bank? [4 pts] B Note: For all problems on this exam, ignore air resistance and use g = 9.80 m/s2 for the acceleration of gravity when necessary. 1. Dimensional Analysis and Vectors [8 pts] a. Newton's Law of Gravitation says that the force F on a mass m1 due to the gravitational influence of another mass m2 at a distance r is F = Gm1m2/r2, where G is a universal constant. Force has dimensions (mass) (length)/(time)2, or in abbreviated form, ML/T2. What are the dimensions of the constant G? [4 pts] b. As shown in the diagram, vector A has magnitude 3.00 units and is perpendicular to vector B, with magnitude 4.00 units. Vector C has magnitude 6.00 units. The angle = 35.0. Find the vector A+B+C. [4 pts] y B A C x C 2. Velocity and Speed [8 pts] a. Two campers plan to start a hike at a trailhead 3.00 km north of their camp. The first camper wants to pick up a map at the visitor center 5.00 km north of camp before starting the hike, and suggests that they meet at the trailhead. If both walk at a constant speed of 3.50 km/h, how long after the first camper leaves should the second camper wait before starting out, so they reach the trailhead at the same time? [4 pts] Visitor Center Trailhead Campsite b. A robot's position is described by: x(t) = 1.00 m (2.00 m/s) t y(t) = 4.00 m + (3.00 m/s2) t2 Find the robot's average velocity vector between t1 = 2.00 s and t2 = 5.00 s. [4 pts] Path of Camper 2 C Path of Camper 1 3. Acceleration [8 pts] a. An elevator starts from rest on the ground floor, accelerates at 1.00 m/s2 until it reaches its maximum speed of 5.00 m/s, travels for some time at this speed, and then eventually decelerates at 1.00 m/s2 to stop at a floor 180.0 m above. How much time does this elevator ride take? [4 pts] b. An object starts from rest at position x = 0 m at time t = 0 s. Five seconds later, at t = 5.0 s, it is observed to be at x = 50.0 m, with velocity vx = 12 m/s. Was the object's acceleration uniform? Explain your reasoning [3 pts], and sketch a velocity vs. time graph consistent with the data given [1 pt]. vx (m/s) t (s) C 4. Motion in Two Dimensions [8 pts] a. An archer shoots an arrow horizontally. The arrow lands 72.0 m away from her, and is stuck into the ground at an angle = 2.50 from horizontal. With what speed was the arrow fired? (You do not need to know the height of the archer, or the distance from the ground that the arrow was fired). [4 pts] Arrow Ground b. Chad throws a pass to his teammate Mario, who is 15.0 m meters in front of Chad and running directly away from him at 7.00 m/s when the pass is released. Chad throws the pass at a 45.0 angle from the horizontal, and Mario catches it without changing his speed or direction. What was the initial speed of Chad's throw? [4 pts] C 5. Circular and Relative Motion [8 pts] a. The minuteshand of a clock is 7.00 cm long. What is the magnitude of the acceleration at a point on the outer end of the hand? [4 pts] b. A rowboat crosses a 140m wide river that is flowing to the east at a speed of 1.00 m/s. The rower can row at a speed of 2.00 m/s relative to the water. If the rower rows straight north, how far downstream will the boat land on the opposite bank? [4 pts] C Note: For all problems on this exam, ignore air resistance and use g = 9.80 m/s2 for the acceleration of gravity when necessary. 1. Dimensional Analysis and Vectors [8 pts] a. Newton's Law of Gravitation says that the force F on a mass m1 due to the gravitational influence of another mass m2 at a distance r is F = Gm1m2/r2, where G is a universal constant. Force has dimensions (mass) (length)/(time)2, or in abbreviated form, ML/T2. What are the dimensions of the constant G? [4 pts] b. As shown in the diagram, vector A has magnitude 3.00 units and is perpendicular to vector B, with magnitude 4.00 units. Vector C has magnitude 6.00 units. The angle = 40.0. Find the vector A+B+C. [4 pts] y B A C x D 2. Velocity and Speed [8 pts] a. Two campers plan to start a hike at a trailhead 3.00 km north of their camp. The first camper wants to pick up a map at the visitor center 5.00 km north of camp before starting the hike, and suggests that they meet at the trailhead. If both walk at a constant speed of 2.50 km/h, how long after the first camper leaves should the second camper wait before starting out, so they reach the trailhead at the same time? [4 pts] Visitor Center Trailhead Campsite b. A robot's position is described by: x(t) = 3.00 m (4.00 m/s) t y(t) = 2.00 m + (1.00 m/s2) t2 Find the robot's average velocity vector between t1 = 2.00 s and t2 = 5.00 s. [4 pts] Path of Camper 2 D Path of Camper 1 3. Acceleration [8 pts] a. An elevator starts from rest on the ground floor, accelerates at 1.00 m/s2 until it reaches its maximum speed of 5.00 m/s, travels for some time at this speed, and then eventually decelerates at 1.00 m/s2 to stop at a floor 160.0 m above. How much time does this elevator ride take? [4 pts] b. An object starts from rest at position x = 0 m at time t = 0 s. Five seconds later, at t = 5.0 s, it is observed to be at x = 42.0 m, with velocity vx = 12 m/s. Was the object's acceleration uniform? Explain your reasoning [3 pts], and sketch a velocity vs. time graph consistent with the data given [1 pt]. vx (m/s) t (s) D 4. Motion in Two Dimensions [8 pts] a. An archer shoots an arrow horizontally. The arrow lands 80.0 m away from her, and is stuck into the ground at an angle = 2.25 from horizontal. With what speed was the arrow fired? (You do not need to know the height of the archer, or the distance from the ground that the arrow was fired). [4 pts] Arrow Ground b. Chad throws a pass to his teammate Mario, who is 15.0 m meters in front of Chad and running directly away from him at 9.00 m/s when the pass is released. Chad throws the pass at a 45.0 angle from the horizontal, and Mario catches it without changing his speed or direction. What was the initial speed of Chad's throw? [4 pts] D 5. Circular and Relative Motion [8 pts] a. The minuteshand of a clock is 8.00 cm long. What is the magnitude of the acceleration at a point on the outer end of the hand? [4 pts] b. A rowboat crosses a 150m wide river that is flowing to the east at a speed of 1.00 m/s. The rower can row at a speed of 2.00 m/s relative to the water. If the rower rows straight north, how far downstream will the boat land on the opposite bank? [4 pts] D ...
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This note was uploaded on 09/08/2008 for the course PHYS 3A taught by Professor Casper during the Fall '07 term at UC Irvine.

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