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 a straight line with an acceleration a(v) = − kv^(3/2), where k is a positive constant and v is its speed. At t = 0, its position is x = 0 and its initial velocity is v₀.

(a) Express the velocity v(t) as an explicit function of time.

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

(c) Find the total distance covered by the particle as t → ∞.

2.  A projectile is launched with speed u at an angle θ to the horizontal. At two instants t₁ and t₂ its velocity makes angles α above and β below the horizontal respectively.

(a) Find the time interval Δt = t₂ − t₁ in terms of u, θ, α, β and g.

(b) Find the horizontal distance covered in that interval.

(c) Show that if α = β, the average velocity over the interval is purely horizontal.

3.  A particle moves in the x-y plane with a velocity field given by v(x,y) = (−ky)î + (kx)ĵ, where k is a positive constant. At t = 0, the particle is located at (R,0).

(a) Show that the trajectory of the particle is a circle of radius R.

(b) Find the position vector r(t) as an explicit function of time.

(c) Compute the acceleration vector a(t) and verify that its magnitude is constant.

4.  Two particles A and B are projected simultaneously from the same point on horizontal ground. Particle A is thrown vertically upward with speed u_A, while particle B is thrown at an angle θ = 60° with speed u_B = 20 m/s. Take g = 10 m/s².

(a) Find u_A such that the relative velocity v_AB is purely horizontal at all times.

(b) Find the distance between the two particles when B reaches its maximum height.

(c) Show that the line joining A and B maintains a fixed direction at all times, and find the value of u_A for which that line makes 45° above the horizontal.

5.  A particle moves in a circular path of radius R = 2 m such that its speed depends on the distance s traveled along the path according to v(s) = b√(s), where b = 2 m^(1/2)/s.

(a) Calculate the tangential acceleration a_(t) as a function of s.

(b) Find the centripetal acceleration a_(c) as a function of s.

(c) Determine the total distance s traveled when the tangential and centripetal accelerations become equal in magnitude.

6.  A particle starts from rest at x = 0 and is subjected to an acceleration a(x) = (k)/(x + x_0), where k = 18 m³/s² and x₀ = 2 m.

(a) Find v(x) as an explicit function of position.

(b) Calculate the velocity of the particle at x = 6 m.

(c) Derive an integral expression for the time t required to reach x = 6 m.

7.  A target is located at coordinates (x_(T),y_(T)) = (120 m,40 m) relative to a launch point on flat ground. A projectile is launched with initial speed u = 50 m/s. Taking g = 10 m/s²:

(a) Formulate the quadratic equation in tan θ required for the projectile to hit the target.

(b) Calculate the two possible launch angles θ₁ and θ₂.

(c) Find the time of flight for both trajectories.

8.  A particle moves along a straight line such that its velocity varies as v(t) = v₀cos (ωt).

(a) Find the maximum displacement x_(max) from the initial position x(0) = 0.

(b) Calculate the total distance covered by the particle between t = 0 and t = (3π)/(2ω).

(c) Determine the average speed over the interval t = 0 to t = (π)/(ω).

9.  A swimmer can swim with a speed v_(s) = 3 m/s in still water. She wants to cross a river of width W = 180 m flowing with current speed v_(r) = 5 m/s.

(a) Find the angle relative to the upstream bank at which she should swim to minimize her downstream drift.

(b) Calculate the minimum drift distance along the opposite bank.

(c) Calculate the time required to complete the crossing along this path.

10.  A particle moves in two dimensions with position coordinates x(t) = asin (ωt) and y(t) = a[1−cos(ωt)], where a and ω are positive constants.

(a) Identify the path of the particle by eliminating t.

(b) Express the magnitude of the velocity vector v(t) as a function of time.

(c) Prove that the acceleration magnitude |a(t)| is constant and find its value.

11.  A stone is dropped into a deep well of depth h = 180 m. The sound of the splash is heard at the top after a time interval T. Taking g = 10 m/s² and the speed of sound v_(s) = 340 m/s:

(a) Calculate the time t₁ taken for the stone to reach the water surface.

(b) Calculate the time t₂ taken for the sound to travel back to the top.

(c) Determine the total measured time interval T.

12.  Two objects 1 and 2 move along the x-axis. Their position functions are x₁(t) = 2t² − 8t + 12 and x₂(t) = − t² + 4t + 3 (in SI units).

(a) Find the relative velocity v₁₂(t) = v₁(t) − v₂(t).

(b) Find the time t at which both objects have the same velocity.

(c) Determine the minimum distance between the two objects.

13.  A projectile is thrown with speed u at an angle θ above a ground inclined at angle ϕ (θ > ϕ).

(a) Derive the expression for the range R measured along the inclined plane.

(b) Find the angle θ (in terms of ϕ) that maximizes the range R along the incline.

(c) Show that the maximum range along the incline is R_max = fracu^2g( 1 + sinphi ).

14.  The velocity of a point moving along a straight line varies with displacement as v(x) = α√(x), where α is a positive constant.

(a) Prove that the acceleration of the point is constant.

(b) Find the average velocity of the point over the distance interval x = 0 to x = S.

(c) Express x(t) as an explicit function of time, assuming x(0) = 0.

15.  A particle moves in a plane such that its tangential acceleration is a_(t) = c, a positive constant, and its centripetal acceleration is a_(c) = k²t⁴, where k is a constant. The particle starts from rest at t = 0.

(a) Express the speed v(t) as a function of time.

(b) Determine the radius of curvature ρ(t) of the path as a function of time.

(c) Express ρ as a function of arc length s.

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.