Q4. A particle’s position is x = 5 + 4t − t (metres, seconds). Find its acceleration. When does it stop, and is it speeding up or slowing down at t = 3 s?
v = 4 − 2t and a = −2 m s (constant). It stops at t = 2 s. At t = 3 s, v = −2 m/s and a = −2 m s: same signs, so it is speeding up (moving back in the negative direction).
Q5. A cyclist’s v-t graph is a straight line from 2 m/s at t = 0 to 8 m/s at t = 6 s. Find the acceleration and the displacement in the 6 s.
a = (8 − 2)/6 = 1 m s. The area is a trapezium: ½ × (2 + 8) × 6 = 30 m.
Q6. A ball is thrown straight up. Taking upwards as positive, what are the signs of its velocity and acceleration (a) on the way up, (b) at the top, (c) on the way down? When is it speeding up?
(a) v > 0, a = −g < 0; (b) v = 0, a = −g; (c) v < 0, a = −g. It speeds up only on the way down, where v and a have the same sign.