Show the working, not just the answer. Several of these have a part that is quicker by structure than by algebra — look for it before you start substituting.

1.  A particle moves along the x-axis with a velocity given by v(t) = 3t² − 12t + 9, where v is in m/s and t is in seconds.

(a) Find the position function x(t), assuming x(0) = 2 m.

(b) Determine the time intervals during which the particle is moving in the negative x-direction.

(c) Calculate the total distance traveled by the particle between t = 0 s and t = 4 s.

2.  The acceleration of a particle moving in a straight line is governed by the relation a(v) = – k√(v), where k is a positive constant and v is the speed. At t = 0, the particle passes the origin (x = 0) with an initial velocity v₀.

(a) Derive an expression for the velocity v(t) as a function of time.

(b) Determine the total time T required for the particle to come to rest.

(c) Find the total distance S traveled before stopping.

3.  A vehicle travels along a straight road. It accelerates uniformly from rest at a rate a₁ = 2 m/s² for time t₁, then continues at a constant speed for time t₂, and finally decelerates uniformly at a rate a₂ = 4 m/s² until it comes to rest.

(a) Given that the total distance covered is 1200 m and the total time taken is 60 s, find t₁, t₂, and the maximum speed v_(max).

(b) Sketch the v-t and a-t graphs for this motion.

4.  A particle’s motion in the x-y plane is described by x(t) = 4cos (2t) and y(t) = 3sin (2t), where x and y are in meters and t is in seconds.

(a) Derive the equation of the trajectory in Cartesian coordinates and identify the shape of the path.

(b) Find the velocity vector v(t) and acceleration vector a(t) at t = (π)/(4)~s.

(c) Prove that the acceleration vector is always directed toward the origin.

5.  A ball is projected from the edge of a cliff of height h = 60 m with an initial speed v₀ = 20 m/s at an angle θ = 30^(∘) above the horizontal. Taking g = 10 m/s²:

(a) Calculate the time taken for the ball to reach the ground.

(b) Determine the horizontal range R measured from the base of the cliff.

(c) Find the magnitude and direction of the velocity vector just before the ball strikes the ground.

6.  A particle starts from rest at the origin (x = 0) at t = 0. Its acceleration varies linearly with position according to a(x) = 4 + 2x, where a is in m/s² and x is in meters.

(a) Express the velocity v as a function of position x.

(b) Find the speed of the particle when x = 3 m.

(c) Calculate the acceleration of the particle at x = 3 m.

7.  A swimmer wishes to cross a 200 m wide river flowing with a uniform speed v_(r) = 4 m/s. The swimmer’s speed in still water is v_(s) = 2 m/s.

(a) Can the swimmer reach a point directly opposite the starting point on the other bank? Explain mathematically.

(b) Find the minimum possible drift along the riverbank and the angle relative to the bank at which the swimmer must head to achieve this minimum drift.

(c) Calculate the crossing time corresponding to this minimum drift angle.

8.  The motion of a point along a line is given by x(t) = t³ − 9t² + 24t + 4 (SI units).

(a) Find the average velocity over the interval from t = 0 s to t = 5 s.

(b) Find the instantaneous velocity and the acceleration at t = 1 s.

(c) Determine the displacement between t = 1 s and t = 3 s, and compare it with the total distance travelled over the same interval.

9.  An aircraft flies horizontally at a height of 2000 m above flat ground with a constant speed of 180 m/s. A payload is dropped from the aircraft. Taking g = 10 m/s² and neglecting air resistance:

(a) Find the time of flight of the payload.

(b) Calculate the horizontal distance traveled by the payload from the point of release.

(c) Determine the speed and angle of impact with the ground.

10.  A particle moves in a circular path of radius R = 4 m. Starting from rest at t = 0, its speed increases at a constant rate a_(t) = 3 m/s².

(a) Write expressions for the tangential acceleration a_(t), centripetal acceleration a_(c)(t), and total acceleration a_(net)(t) as functions of time.

(b) Calculate the magnitude of the total acceleration at t = 2 s.

(c) Find the angle ϕ between the total acceleration vector and the velocity vector at t = 2 s.

11.  Two trains A and B are moving on parallel tracks in the same direction. Train A (120 m long) travels at 72 km/h, and Train B (180 m long) travels at 108 km/h.

(a) Convert both speeds to m/s and determine the relative velocity of Train B with respect to Train A.

(b) Calculate the time taken for Train B to completely overtake Train A.

(c) Find the total distance traveled by Train B during this overtaking process.

12.  A particle’s acceleration is given by a(t) = 6t − 24 m/s². At t = 0 its position is x(0) = 5 m and its velocity is v(0) = 36 m/s.

(a) Derive the expressions for the velocity v(t) and the position x(t).

(b) Find the minimum velocity reached by the particle and the time at which it occurs.

(c) Determine the position of the particle at the instant its acceleration is zero.

13.  A stone is dropped from the top of a cliff. One second later, a second stone is thrown vertically downward from the same point with a speed of 20 m/s. Taking g = 10 m/s²:

(a) Find the time t (measured from the drop of the first stone) at which the second stone catches up with the first.

(b) Determine the depth below the top of the cliff where the meeting occurs.

(c) Calculate the speeds of both stones at the instant they meet.

14.  A flexible tape wound on a reel unwinds as a mass falls under gravity. The velocity of a point on the line is governed by v(x) = √(2gx + v_0)^2.

(a) Show that the acceleration of the mass is constant and equal to g.

(b) If v₀ = 2 m/s, find the distance required to double the speed.

(c) Plot the expected v² versus x relationship.

15.  A projectile is launched with speed u at an angle θ above the horizontal over level ground.

(a) Derive the parametric equations for position x(t) and y(t), and eliminate t to obtain the trajectory equation y(x).

(b) Derive expressions for the total time of flight T, maximum height H, and horizontal range R.

(c) Prove that the maximum horizontal range occurs at θ = 45^(∘).

16.  A particle undergoes linear motion with piecewise acceleration: a(t) = beginmatrix 4~m/s^2 & 0 leq t leq 3~s 0 & 3 < t leq 7~s

2~m/s^2 & 7 < t leq 11~s endmatrix . Assuming the particle starts from rest at x(0) = 0:

(a) Calculate the velocity at t = 3 s, 7 s, and 11 s.

(b) Calculate the final displacement x(11 s).

(c) Construct the complete velocity-time table and compute average speed over the 11 s interval.

17.  A motorboat heading due north across a river at 6 m/s relative to the water encounters a current flowing east at 2.5 m/s. The width of the river is 300 m.

(a) Determine the magnitude and direction of the resultant velocity of the boat relative to the riverbank.

(b) Calculate the time needed to reach the opposite bank.

(c) Find the downstream distance (drift) where the boat lands.

18.  A particle moves in a circle of radius R = 2 m such that its distance along the arc from a fixed point is given by s(t) = t³ − 3t² m.

(a) Find the tangential velocity v(t) and tangential acceleration a_(t)(t).

(b) Find the centripetal acceleration a_(c)(t) at t = 2 s.

(c) Determine the magnitude of the total acceleration vector at t = 2 s.

19.  A point A moves along a straight line with uniform speed v_(A) = 10 m/s. Point B starts from rest at the same location at t = 0 and pursues A along the same line with constant acceleration a_(B) = 2 m/s².

(a) Write position equations x_(A)(t) and x_(B)(t).

(b) Find the time t at which B catches up with A.

(c) Calculate the distance traveled by both points up to the catching point, and the speed of B at that instant.

20.  An elevator cabin of height h = 2.7 m begins ascending with a constant acceleration a = 1.2 m/s². At t = 2 s after start, a bolt drops loose from the ceiling of the elevator. Taking g = 9.8 m/s²:

(a) Calculate the acceleration of the bolt relative to the elevator floor.

(b) Determine the time taken for the bolt to hit the floor of the elevator.

(c) Calculate the net displacement of the bolt relative to the shaft ground frame during its fall.

If your answer isn’t there

No answers are given, deliberately — and no question tells you which part it comes from, because deciding that is most of the work. If one defeats you, K16 maps twelve physical questions to the tool each one needs. Working it out a second time teaches more than checking a key.

But if you have worked carefully and your answer matches none of the options, it could be our mistake rather than yours. Every question was checked before publication, and checking is not the same as being right.

Those are the questions worth staying with. Rework them, and argue them out with a friend or a teacher. Showing that none of four options can be right is harder physics than picking the one that is — it needs you to trust your own derivation rather than search a menu for something familiar. A student who can do that has understood the motion.

Then tell us, and we will look at it properly. If the question is wrong we will correct it and say so on this page.