Displacement & velocity time graphs

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1 Displacement & elocity time graphs Model Answers 1 Leel Exam Board Subject Module Topic A LEVEL Edexcel Mathematics Mechanics & Statistics Constant acceleration Sub-Topic Booklet Model Answers 1 Displacement & elocity time graphs Time Allowed: Score: Percentage: 35 minutes /9 /100 Grade Boundaries: A* A B C D E U >85% 77.5% 70% 6.5% 57.5% 45% <45% Model Answer Key Red = Answer - This is what you need to write to get the mark - Each bullet point represents 1 mark Blue = Explanation - This is here to help you understand the answer - You DON'T need to write this to get the marks 1 Model answers are copyright. 018 Sae My Exams Ltd. All rights resered.

2 1. 1 (m s ) 30 O T t (s) Figure 4 The elocity-time graph in Figure 4 represents the journey of a train P traelling along a straight horizontal track between two stations which are 1.5 km apart. The train P leaes the first station, accelerating uniformly from rest for 300 m until it reaches a speed of 30 m s 1. The train then maintains this speed for T seconds before decelerating uniformly at 1.5 m s, coming to rest at the next station. (a) Find the acceleration of P during the first 300 m of its journey. Because the train has constant acceleration, we can use the SUVAT equation = uu + aaff The initial elocity of the train is 0 as it starts from rest. It traels s=300m until it reaches elocity = = aa(300) aa = a = 1.5 ms Model answers are copyright. 018 Sae My Exams Ltd. All rights resered.

3 (b) Find the alue of T. (5) The distance that the train traels is the area under the speed-time graph. The train traels a total distance of 1.5km (1500m), so this must be the total area underneath the graph. To find the time taken for the train to accelerate use the SUVAT equation = uu + aamm 30 = mm substitute in the known alues mm = 0FF To find the time taken for the train to decelerate use the SUVAT equation = uu + aamm 0 = mm substitute in the known alues mm = 4FF The distance traelled is the area under the graph. The area of a trapezium is 1 h (aa + bb) We know that the train traels 1500m oerall, so we can substitute in alues 1 30 (0 + TT TT) = TT = 100simplify TT = 56 3 T= 8s Model answers are copyright. 018 Sae My Exams Ltd. All rights resered.

4 A second train Q completes the same journey in the same total time. The train leaes the first station, accelerating uniformly from rest until it reaches a speed of V m s 1 and then immediately decelerates uniformly until it comes to rest at the next station. (c) Sketch on the diagram aboe, a elocity-time graph which represents the journey of train Q. Train Q is represented on the below graph in red. The key features are that the same oerall time is taken for train Q to go between the two stations, and that the shape of the graph is a triangle. Notice that the elocity V must be greater than =30. (d) Find the alue of V. (6) Train Q traels the same distance as train P, therefore the area under the graph of Q must also equal 1500m. The area is simply a triangle. AAbbmmaa = 1 VV 7 = 1500 V = 15 3 ms 1 (Total 15 marks) 4 Model answers are copyright. 018 Sae My Exams Ltd. All rights resered.

5 . A car is traelling along a straight horizontal road. The car takes 10 s to trael between two sets of traffic lights which are 145 m apart. The car starts from rest at the first set of traffic lights and moes with constant acceleration for 30 s until its speed is m s 1. The car maintains this speed for T seconds. The car then moes with constant deceleration, coming to rest at the second set of traffic lights. (a) Sketch, in the space below, a speed-time graph for the motion of the car between the two sets of traffic lights. ms s (T + 30) s t (b) Find the alue of T. (3) The area under the graph is the distance traelled which is known. Hence 145 = TT + 1 (10 TT 30) 145 = 11 (30 + TT + 90 TT) 195 = TT + 10 TT = 7711s 5 Model answers are copyright. 018 Sae My Exams Ltd. All rights resered.

6 A motorcycle leaes the first set of traffic lights 10 s after the car has left the first set of traffic lights. The motorcycle moes from rest with constant acceleration, a m s, and passes the car at the point A which is 990 m from the first set of traffic lights. When the motorcycle passes the car, the car is moing with speed m s 1. (c) Find the time it takes for the motorcycle to moe from the first set of traffic lights to the point A. (4) Motorcycle moing with acceleration a ms -1 t 10 s t s 30 s (T + 30) s The motorcycle reaches point A at time t. We can calculate the area under the graph for the distance as before (using the car) 990 = (tt 30) tt = tt = 60 This is in the car s time frame. The bike leaes 10s later so the time the takes is = 50s (d) Find the alue of a. Using SUVAT 99 = uutt + 77 mmtt = aa 50 aa = = 0.79ms - (Total 11 marks) 6 Model answers are copyright. 018 Sae My Exams Ltd. All rights resered.

7 3. A ball of mass 0.3 kg is released from rest at a point which is m aboe horizontal ground. The ball moes freely under graity. After striking the ground, the ball rebounds ertically and rises to a maximum height of 1.5 m aboe the ground, before falling to the ground again. The ball is modelled as a particle. (d) Sketch, in the space proided, a elocity-time graph for the motion of the ball from the instant when it is released until the instant when it strikes the ground for the second time. (3) Velocity-time graph: Maximum elocity at the initial point The ball strikes the ground u Acceleration under g = -9.8 m/s = the gradient of the line We fix the direction up as positie. The ball reaches the highest point aft er rising the second time and then falls. Its elocity is lower than the one in t he initial displacement -u t Acceleration under g. This line should be just as steeper as the acceleration represented aboe, since the graitational acceleration is the same. When the ball starts at time t = 0, its elocity is maximum,. As it goes down ertically, the ball accelerates due to its weight acting downwards, W = mg. At the point = 0 m/s, the ball is at its highest point. The negatie elocity represents that the ball is moing in the negatie direction, downwards, and speeding up. (Total 3 marks) 7 Model answers are copyright. 018 Sae My Exams Ltd. All rights resered.

d / cm t 2 / s 2 Fig. 3.1

d / cm t 2 / s 2 Fig. 3.1 7 5 A student has been asked to determine the linear acceleration of a toy car as it moves down a slope. He sets up the apparatus as shown in Fig. 3.1. d Fig. 3.1 The time t to move from rest through a