2. A homemade car is capable of accelerating from rest to 100 km hr 1 in just 3.5 s. Assuming constant acceleration, find:

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1 Preliminary Work 1. A motorcycle accelerates uniformly from rest to a speed of 100 km hr 1 in 5 s. Find: (a) its acceleration (b) the distance travelled in that time. [ Answer: (i) a = 5.56 ms 2 (ii) x = 69.5 m ] 2. A homemade car is capable of accelerating from rest to 100 km hr 1 in just 3.5 s. Assuming constant acceleration, find: (a) its acceleration (b) the distance travelled in that time. [ Answer: (i) a = 7.94 ms 2 (ii) x = 48.6 m ] 3. A car moves along a straight roadway with constant speed of 40 km/hr. Find: (a) the increase in speed if the car accelerates at 5 km/hr 2 for 50 min (b) the distance traveled by the car during that time (c) the stopping time and distance if the brakes are applied, causing deceleration of 10 km/hr 2. [ Answer: (i) v = 44.2 km/hr (ii) x = 35.1 km (iii) x = 97.9 km, t = 4.42 hr ] 4. A tennis ball is thrown vertically upwards with a velocity of 15 m s 1. Find the maximum height reached. How long does the ball remain in the air? [ Answer: (i) h = 11.5 m (ii) T = 3.1 s ] 5. A stone is dropped from a height of 100 m. How many seconds does it take for the stone to reach the ground? What is the speed of the stone just before striking the ground? [ Answer: (i) t = 4.52 s (ii) 44.3 ms 1 ] 6. The figure shows a graph of distance s travelled against time for an aircraft on a runway, accelerating towards take-off. (a) Calculate the average speed between times t = 5 s and t = 15 s. (b) Use the graph to estimate the instantaneous speeds at the same two times. (c) Estimate the average acceleration between t = 5 s and t = 15 s. [ Answer: 25 m/s; 12.5 m/s, 37.5 m/s; 2.27 m/s 2 ] 1

2 Tute M2 : LINEAR MOTION UNDER CONSTANT ACCELERATION 1 1. A funny car can cover 400 m from a resting start in only 4.9 s. After passing the 600 m mark, this funny car then decelerates uniformly to rest (using both brakes and a parachute) in a distance of only 300 m. Calculate: (a) its acceleration before the 400 m mark. (b) its velocity at the 400 m mark. (c) the deceleration during its second part of the journey. [ Answer: (a) a = ms 2 (b) v = ms 1 (c) a = ms 2 ] 2. A spider is moving around in a room. It will move along the vectors OA, AB, BC and CD. It will take the spider 1/2hr to reach D. An air-conditioning unit allows a breeze to flow continuously with a force of 0.5 N, aimed along the direction of 45 above the horizontal in the xy plane. y (a) What is the spider s average velocity? (b) What is the average speed of the spider? air-cond D 45 o A (3,1,1) (c) What is the total work done by the spider against the wind? 1m 3m O B x z C 3m [ Answer: (a) (0.56j+1.67k) 10 4 m/s (b) m/s (c) 0.35 J] 3. A stone is thrown vertically upwards at 15 m s 1 from the surface of the Moon. Given that the acceleration due to gravity on the Moon is 1.62 m s 2, calculate 1 September 18,

3 (a) the maximum height reached (b) the total time the stone remains in flight (c) the initial vertical velocity required to make the stone reach twice the height found in part (a). [ Answer: (a) 69.4 m (b) 18.5 s (c) 21.2 m s 1 ] 4. A rock is dropped from an overhead bridge 30 m above a freeway. The rock strikes the driver of a car travelling at 100 km hr 1. With what speed does the rock hit the driver? [ Answer: (i) v = 36.9 ms 1 = km hr 1 ] 5. A vehicle travelling at a speed of 10 m s 1 undergoes an acceleration of 2 m s 2 for 3 seconds and then an acceleration of 3 m s 2 for the next 4 seconds. Find: (a) its final velocity (b) the distance travelled after 7 seconds (c) the distance travelled during the third second. [ Answer: (i) v f = 28 ms 1 (ii) x = 127 m (iii) 15 m ] 6. A stuntman steps off the top of a building 100 m tall and successfully lands in the middle of a pile of mattresses 2 m high. (a) What is the velocity of the stuntman just before hitting the mattresses? (b) If the mattresses are compressed by 1.5 m before stopping the stuntman, what is the deceleration experienced by the stuntman? [ Answer: (a) 43.8 m s 1 (b) m s 2 65g ] 3

4 7. A parachutist is descending with a constant speed of 10 m s 1. When he is 50 m above the ground he loses his shoes and it falls to the ground. How long after the shoe hits the ground does the parachutist touch down? [ Answer: 2.68 s ] 8. Two cars approach each other in opposite directions along a straight road. At time t = 0, both cars are 300 m apart. Car A is travelling at 60 km/hr while car B is travelling at 50 km/hr. Find the time and positions of the cars when they meet if, (a) both cars continue to move at constant speed (b) car A continues to move at constant speed while car B starts to accelerate at 1.6 m/s 2 [ Answer: (i) t = 9.8 s, s A = 164 m, s B = 136 m (ii) t = 8.09 s, s A = 135 m, s B = 165 m ] 9. The velocity of a toy train on a straight track is graphed below for the indicated time intervals I, II, III and IV. Calculate the following: (a) acceleration of train in I (b) distance travelled in I (c) distance travelled in II (d) acceleration in III (e) distance travelled in III (f) acceleration in IV (g) distance travelled in IV (h) total distance travelled. [ Answer: (i) a I = 1.5 m s 2 (ii) d I = 12 m (iii) d II = 36 m (iv) a III = 0.8 m s 2 (v) d III = 40 m (vi) a IV = 1.67 m s 2 (vii) d IV = 30 m (viii) D = 118 m ] 4

5 10. Ball A is dropped from a roof of a 25 m high building. At the same time, ball B is projected upward, from the ground, with a velocity of 15 m/s. A third ball C is also dropped from the roof 1 s after the first ball is dropped. (a) Find the positions of ball A and ball B when they pass each other. (b) Find the position and velocity of the ball C when the ball A hits the ground. [ Answer: (i) m above ground level (ii) m above ground level, at m/s ] 11. Past Exam Question Bob is standing on the roof of a tall building and drops a tennis ball directly downward over the side of the building. Mary, from inside the building, notices that it takes the ball 0.1 s to pass by her 1 m high window. She continues timing and measures a further 6,0 s for the ball to reappear at the base of her window after bouncing up from the ground. From what height was the ball initially released by Bob? Assume the ball bounces elastically and take the acceleration due to gravity to be g = 10 m/s 2. [ Answer: m ] 12. Student A throws a ball up the side of a building with an initial velocity of 45 m/s. Student B, standing on the roof of the building, catches the ball just as it reaches the roof 5 s later. In the meantime, Student C, being inside the building, observes that the ball takes 0.05 s to pass his 2 m high window. Find (a) the height of the building (b) the time it takes for the ball to reach the bottom of the window (c) how high the window is from the ground level [ Answer: 100 m ; s ; 20.2 m ] 5

6 Further Practice Problems: 1. A particle initially located at the origin has an acceleration of a = 3.00j m/s and an initial velocity of v = 5.00i m/s. Find (a) the vector position and velocity at any time t and (b) the coordinates and speed of the particle at t = 2.00 s. [ Answer: r = 5ti + 1.5t 2 j ; v = 5i + 3tj ] 2. Speedy Sue, driving at 30.0 m/s, enters a one-lane tunnel. She then observes a slowmoving van 155 m ahead travelling at 5.00 m/s. Sue applies her brakes but can only accelerate at 2.00 m/s 2 because the road is wet. Will there be a collision? State how you decide? [Hint: Find a expression for the position of the car and the position of the van as a function of time.] If yes, determine how far into the tunnel and at what time the collision occurs. If no, determine the distance of closest approach between Sues car and the van. [ Answer: Collide with truck at t = 11.4 s ] 3. A moving car decelerates for 5 s and comes to a complete stop. It travels 75 m in the process. Determine its initial value of speed and its rate of deceleration. [ Answer: 30 m/s ; 6 m/s 2 ] 4. A stone is thrown upward from 10 m above the ground, with an initial speed of 15 m/s. Find the maximum height attained by the ball. How much time does it take to reach the maximum height? What is the velocity of the ball when it reaches its initial position? [ Answer: m above the starting point ; 1.5 s ; 15 m/s ] 5. A tennis ball is dropped from 1.2 m above the ground. It rebounds to a height of 1 m. With what velocity does it hit the ground? With what velocity does it leave the ground? If the ball were in contact with the ground for 0.01 s, find its acceleration while touching the ground. 6

7 [ Answer: 4.9 m/s ; 4.5 m/s ; 940 m/s 2 ] 6. A car goes down a certain road at an average speed of 40 km/h and returns along the same road at an average speed of 60 km/h. Calculate the average speed for the round trip. [ Answer: 48km/h ] 7. Emily takes a trip, driving with a constant velocity of 90 km/h to the north except for a 30 min rest stop. If Emily s average velocity is 75 km/h to the north, how long does the trip take? [ Answer: 3 hrs, 225 km ] 8. You are driving towards a traffic signal when it turns yellow. Your speed is 55 km/h, and your best deceleration has the magnitude of 5.15 m/s 2. Your best reaction time before breaking is 0.75 s. To avoid having the front of your car enter the intersection after the light turns red, should you (i) break to a stop or (ii) continue to move at 55 km/h if the distance to the intersection and the duration of the yellow light are (a) 30m and 2.8s, and (b) 32m and 1.8s? [ Answer: (a) he will cross intersection if he keeps at constant speed, (b) he will not be able to stop or cross the intersection ] 9. An automobile traveling 90 km/h overtakes a 1.5 km long train traveling in the same direction on a track parallel to the road. If the train s speed is 70 km/h, find how long does it take the car to pass it, and how far will the car have traveled in the time? [ Answer: 4.5 min, 6.75 km ] 7

8 An example of non uniform acceleration: 1. A charged particle in an electric field moves so that its displacement r relative to the origin of the (x, y, z) axes is given at any time t by r = t i t 3 j m. (a) Find the velocity v and acceleration a of the particle at any time t. (b) Find the initial velocity v 0. (c) Sketch r, v and a at t = 1 s. (d) Sketch the trajectory of the particle. [ Answer: (i) v = i-3t 2 j ms 1 ; a = -6tj ms 2 (ii) v o = i ms 1 ] 8