Mechanics 1. Motion MEI, 20/10/08 1/5. Chapter Assessment

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Erica Robbins
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1 Chapter Assessment Motion. A snail moving across the lawn for her evening constitutional crawl is attracted to a live wire. On reaching the wire her speed increases at a constant rate and it doubles from. ms - in ten. She remains at this speed for a further while she remains in contact with the wire. a. Draw a velocity time graph to represent this situation when the snail is in contact with the wire. [] b. Calculate the acceleration. [] c. Calculate the distance travelled during this time. []. A particle is constrained to move along a straight line through O. It starts initially at D, which is a fixed distance from O in the positive direction and moves at a constant velocity of ms - for until it is m from O. It remains m from O for a further, after which it travels in the opposite direction for. The total distance travelled is 8 m. a. Draw a distance-time graph to represent this situation. [] b. What is the velocity in the final stage of the journey? [] c. How far from O does the particle stop? [] d. Draw a velocity time graph. [] e. What is the final displacement of the particle from its initial position? []. A, B and C are three points lying in that order on a straight road with AB km and BC km. A man runs from A to B at kmh - and then walks from B to C at 8 kmh -. Find: a. the total time taken to travel from A to C [] b. the average speed of the man from A to C. []. A particle travels in a straight line. The motion is modelled by the v-t diagram below. velocity in metres per second v t time in a. Calculate the acceleration of the particle in the part of the motion from t to t. [] MEI, //8 /

2 b. Calculate the displacement of the particle from its position when t to its position when t 6. [] c. Calculate the displacement of the particle from its position when t to its position when t 7. [] d. Describe the motion of the particle during the interval t 7. []. Acceleration in ms - a 6 7 t - A car is travelling due east along a straight road when it passes a point P. The acceleration of the car during the next 7 is modelled in the accelerationtime graph above, where a ms - is the acceleration of the car due east and t is the time after passing the point P. a. Explain why the speed of the car is greatest when t 6. [] The speed of the car when it passes P is ms -. b. Calculate the speed of the car when t. [] c. Show that, when t, the speed of the car is ms -. [] d. Show that, for t 7, the acceleration is given by a -t +. [] e. Explain how the graph may be used to show that the speeds at t and t 7 are equal. [] Total: marks MEI, //8 /

3 Solutions to Chapter Assessment. (i) Velocity in ms -.. change in velocity (ii) Acceleration time... The acceleration is. ms -. (iii) Distance travelled area under graph Area under graph from t to t is (. +.). Area under graph from t to t is.. Total area. Distance travelled. m or. cm.. (i) at a speed of ms - gives a distance of 6 m Distance in metres Remember: distance is distance travelled. It is not the same as position, which is location relative to the origin, or displacement, which is position relative to the starting point. 6 MEI, //8 /

4 displacement The displacement is (ii) Velocity negative as the motion time is towards the origin 6 The velocity in the last part of the motion is -6 ms -. (iii) After, the particle is m from O. It then travels in the opposite direction for m, so it stops m from O. (iv) Velocity in ms (v) After the first, the particle has moved 6 m and is m from O. So its initial position is m from O. From part (iii), its final position is m the other side of O. Displacement final position - initial position 6 Displacement -6 m.. (i) Time from A to B. Time from B to C. 8 Total time taken to travel from A to C.7 hours minutes total distance (ii) Average speed total time 9.7 Average speed kmh -. MEI, //8 /

5 change in velocity. (i) Acceleration time taken - ms (ii) Displacement area under graph ( ) ( ( ) ) ( ) m (iii) Displacement between t 6 and t 7 is Displacement between t and t 7 is 7 6 m. (iv) The particle decelerates at a constant rate of ms -. When t 6 the particle is instantaneously at rest and thereafter the particle is travelling in the opposite direction.. (i) The acceleration is positive until t 6, so the speed is increasing from t to t 6. At t 6 the acceleration is zero, and then it is negative, so the speed then begins to decrease. (ii) Increase in speed area under graph between t and t Increase in speed 6 Speed when t ms -. (iii) Increase in speed between t and t is area under graph Increase in speed ( + ) 7 Speed at t is ms -. (iv) Gradient of graph for t 7 - When t 6, a (a ) -(t 6) a -t + (v) The area under the graph for t 6 is equal to the area under the graph for 6 t 7, but of opposite sign. Therefore the total change in velocity is zero, so the speeds at t and t 7 are equal. MEI, //8 /

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Created by T. Madas CALCULUS KINEMATICS. Created by T. Madas CALCULUS KINEMATICS CALCULUS KINEMATICS IN SCALAR FORM Question (**) A particle P is moving on the x axis and its acceleration a ms, t seconds after a given instant, is given by a = 6t 8, t 0. The particle