Motion graphs practice

1. The plank has one steady slope, so gravity's pull along it never changes: a is constant and, pointing the way the ball rolls, positive. A flat line above the axis.
2. The ball starts at rest and gains the same amount of speed each second: v is a straight line that starts at 0 and rises.
3. Each second it covers more ground than the second before: d starts flat and bends upward.

Position instead of distance: if the ball moves in your positive direction, position looks exactly like the distance graph here. If you pick the other direction as positive, every graph flips below the axis but keeps its shape.

1. Each straight piece has its own constant pull, so a is two flat steps: a low positive value on the gentle part, then a higher one on the steep part.
2. v is a straight line on each piece. It rises slowly, then more steeply after the bend. The ball does not suddenly change speed at the bend, so v has a corner, not a jump.
3. d bends upward the whole time and bends up even faster on the steep part. It never goes down.

1. While it climbs, gravity pulls back down the slope, against the motion: a is constant and negative the whole time, even at the moment it stops.
2. The ball loses speed at a steady rate: v is a straight line sloping down, reaching 0 where the ball stops for an instant.
3. d rises but levels off as the ball slows, and is flat for an instant where it stops.
4. Dashed part: as it rolls back, v carries on along the same straight line below the axis (the ball now moves the other way), a does not change, and d keeps rising, because distance only adds up.

Position version (up the ramp = +): position rises, is flat at the top, then comes back down the same way: a hill shape.

1. Three pieces, so three flat steps of a: negative on the steep climb (it slows quickly), zero on the flat top, and a small positive value on the gentle way down.
2. v drops quickly in a straight line, stays level across the top, then rises slowly. It never reaches 0, so the car never stops.
3. d always rises: it levels off a little while the car slows, is a straight line across the top, and bends upward again on the way down.
4. No friction: it finishes at the same height it started, so it ends with the same speed it started with.

1. In words: the cart speeds up down the steep left slope, rolls across the bottom at a steady speed, slows down as it climbs the right slope, stops for an instant at the same height it started (no friction), and then rolls back (dashed).
2. a (measured along the way it first moves): positive and large on the steep slope, zero on the flat, negative and smaller on the gentler slope.
3. v: rises from 0 in a straight line, stays level, then falls back to 0 more slowly than it rose. Dashed: it continues below the axis as the cart comes back.
4. d: bends upward, then a straight line, then levels off at the stop, then keeps rising on the way back.

Position version (right = +): the cart moves right first, so position rises (bending up, then straight, then levelling off) and comes back down on the return.

1. Start to B: it starts from rest and speeds up going forward (v positive and rising, so a is positive).
2. B to C: it moves forward at a steady velocity (v level, a = 0).
3. C to D: still going forward, but slowing down, and it stops at D (v falls to 0, a negative and twice as big as at the start, because this slope is twice as steep).
4. D to E: it speeds up going backward. v is now negative and growing in size, and a is still negative: the same straight line carries on through zero.
5. E to F: it moves backward at a steady velocity (a = 0).
6. F to G: it slows down while still going backward and stops at G (v rises back to 0, a positive).
7. The distance graph keeps rising the whole time; it is only flat for an instant at D and at G, where the robot stops.

Where does it end? The area above the axis (forward) is 10.5 units and the area below (backward) is 7.5 units, so it finishes 3 units ahead of where it started.