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Nov 2005 p4 q5
3718
The diagram shows the displacement-time graph for a car’s journey. The graph consists of two curved parts AB and CD, and a straight line BC. The line BC is a tangent to the curve AB at B and a tangent to the curve CD at C. The gradient of the curves at t = 0 and t = 600 is zero, and the acceleration of the car is constant for 0 < t < 80 and for 560 < t < 600. The displacement of the car is 400 m when t = 80.
Sketch the velocity-time graph for the journey.
Find the velocity at t = 80.
Find the total distance for the journey.
Find the acceleration of the car for 0 < t < 80.
Solution
(i) The velocity-time graph consists of three straight line segments: a positive gradient for 0 < t < 80, a zero gradient for 80 < t < 560, and a negative gradient for 560 < t < 600. The graph starts and ends on the t-axis at t = 0 and t = 600, respectively.
(ii) Using the area under the velocity-time graph for 0 < t < 80, we have:
\(\frac{1}{2} \times 80 \times v = 400\)
Solving for \(v\), we get \(v = 10 \text{ ms}^{-1}\).
(iii) The total distance is the area under the velocity-time graph. The area of the trapezium is: