Applications of Calculus  Graphs of motion in one direction.
Test Yourself 1.
The questions on this page focus on: 
1. given a displacement graph. 
2. given a velocity graph. 
3. given an acceleration graph. 
Given displacement graph.  1.
A particle moves along a straight line for 7 seconds starting at a fixed point S at time t = 0. The graph of x(t) denoting the position of P at time t is shown above. Answers.(i) Particle started at S = 0.5 (approx). (ii) In a negative direction as the gradient is negative. (iii) At t = 1 and t = 4. (iv) At t = 4 seconds. (v) At t = 5 the gradient is negative and the concavity is still negative. So the particle is speeding up in a negative direction  so slowing down in a positive direction. 
2. The graph showing the motion of a particle over the first 3 seconds is shown below (displacement in metres).


3.  
4. Sketch a graph of displacement against time given that, during the period of motion, the displacement is always negative, the velocity is always negative and the acceleration is always positive.  
5. Sketch a graph of displacement against time given that, during the period of motion, the displacement is always positive, the velocity is always positive and the acceleration is always negative.  
Given velocity graph.  6. The diagram below describes the velocity of a particle (in m/sec) over the first 4 seconds.
Answers.(i) 20 m/sec. (ii) Acc = 10 m/sec^{2}. (iii) At t = 2. (iv) Displacement = 10 m. 
7.
The graph above shows the velocity of a particle against time of travel. Initially the particle was at the origin. Answers.(i) Furthest at 4 secs. (ii) After 7 secs, 4 m to right of origin. 

8. The graph shows the velocity v (in metres/sec) of a particle, moving from the origin, as a function of time t.
Answers.(i) Max acceleration at t = 3. (ii) Distance = 240 m. (iii) Returns to origin at t = 18 secs. 

9.
The diagram above shows the velocitytime graph for a particle moving in a straight line. State the times between t = 0 and t = 8 at which: Answers.(i) v = 0 at t = 1 and 5 secs. (ii) t = 3 and 7. (iii) 1 < t < 3 (v & a both positive) and 5 < t < 7 (v & a both negative). 

10. An object is moving along in a straight line. The graph below shows the velocity (in m/sec), , of the object as a function of time t.
The coordinates of the points on the graph are A (1, 0), The velocity is constant for t ≥ 5.


11.
The diagram above shows the velocitytime graph for a particle moving in a straight line.


12. Below is the graph of a particle’s velocity v in m/sec plotted against time t in seconds. The particle’s motion lasts 10 seconds. (i) State the time(s) when the particle is at rest. 

Given acceleration graph.  13. The graph shows the acceleration of a particle as a function of time t for the period 0 ≤ t ≤ 5.

14.
A particle moves along the xaxis. Initially it is at rest at the origin. The graph shows the acceleration a of the particle as a function of time 0 ≤ t ≤ 5. Answer.(i) Max velocity at t = 2. (ii) Final velocity is 3 m/sec. (iii) Furthest from origin at t = 4. 

15. 