An object's velocity and acceleration have the same sign. What is happening to its speed?
You've already seen the sign rule: an object speeds up when its velocity and acceleration have the same sign and slows down when they have opposite signs. On a velocity-time graph you can make the same call by eye, with no signs to juggle.
A harbor car makes an out-and-back run on a straight waterfront track. Heading out, the positive direction, it climbs from 2 m/s up to 7 m/s over 3 s, then eases down to 1 m/s as it nears the far stop. The run home is the same ride in the negative direction: −2 m/s to −7 m/s, then easing back to −1 m/s.
The car's speed is the size of its velocity: 7 m/s out and −7 m/s home are the same speed. And on the graph, that size is a distance you can see: the distance between the line and the time axis.
Follow the out run, above the axis. From 2 m/s to 7 m/s the line climbs away from the axis, so the car speeds up. From 7 m/s down to 1 m/s the line drops toward the axis, so the car slows down.
Now the run home, below the axis. From −2 m/s to −7 m/s the line falls away from the axis. The speed grows from 2 m/s to 7 m/s, so the car speeds up. The line is falling, and the car is speeding up.
The graph shows the velocity of a kayak. Between t = 0 and t = 6 s, is the kayak speeding up or slowing down?
The line lies below the time axis and rises toward it.
Moving toward the axis means the distance between the line and the axis is shrinking, so the speed is shrinking: from 9 m/s to 3 m/s.
The kayak is slowing down.
The graph shows the velocity of a scooter between t = 0 and t = 5 s. Over that interval, what is the scooter doing?
The graph shows the velocity of a go-kart. Between t = 0 and t = 4 s, is the go-kart speeding up or slowing down?
The line lies above the time axis and rises away from it.
Moving away from the axis means the speed is growing: from 2 m/s to 10 m/s.
The go-kart is speeding up.
Put those two side by side. The kayak's line rose, and the kayak slowed down. The go-kart's line rose, and the go-kart sped up. 'Rising means speeding up' cannot survive that pair. Ask instead where the line is heading: away from the axis, or toward it.
The graph shows the velocity of a rowboat between t = 0 and t = 4 s. Over that interval, what is the rowboat doing?
The graph shows the velocity of a golf cart between t = 0 and t = 6 s. Over that interval, what is the cart doing?
The graph shows the velocity of a delivery van over 8 s. During which interval is the van slowing down?
The graph shows the velocity of a fishing boat between t = 3 s and 7 s. Over that interval, what is the boat doing?
The graph shows the velocity of a remote submarine over 8 s. During which interval is the submarine speeding up?