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. Its velocity-position graph (v-x) is a straight line passing through (0,20 m/s) and (40 m,0 m).

(a) Write the mathematical equation for velocity v as a function of position x.

(b) Determine the expression for acceleration a as a function of x.

(c) Find the acceleration of the particle when it is at x = 10 m.

2.  A ball is projected vertically upward from the top of a tower of height H = 100 m with an initial velocity u = 20 m/s. Simultaneously, another ball is projected vertically upward from the base of the tower with an initial velocity v₀ = 40 m/s. Taking g = 10 m/s²:

(a) Calculate the time t at which the two balls cross each other.

(b) Find the height above the ground at which they cross.

(c) Determine the relative velocity of the second ball with respect to the first ball at the moment of crossing.

3.  The position vector of a particle moving in the x–y plane is r(t) = (t³ − 3t)i + (t² − 2t)j, where r is in metres and t in seconds (t ≥ 0).

(a) Find the velocity vector v(t) and the acceleration vector a(t).

(b) Determine the instant(s) at which the particle moves parallel to the y-axis.

(c) Show that the particle comes momentarily to rest, find the instant at which this happens, and determine its acceleration at that instant.

4.  A river of width W = 600 m flows with a speed v_(r) = 5 m/s. A motorboat can move with a speed v_(b) = 10 m/s in still water.

(a) Calculate the shortest time required to cross the river and the corresponding downstream drift.

(b) Calculate the angle relative to the riverbank at which the boat must be steered to cross the river along the shortest path (zero drift).

(c) Find the time required to cross along this shortest path.

5.  A projectile is launched from ground level with speed u at an angle θ to the horizontal. It just clears two vertical walls of equal height h, separated by a horizontal distance d.

(a) Show that the sum of the times t₁ and t₂ at which the projectile passes the two walls is equal to the total time of flight T.

(b) Derive an expression for the height h of the walls in terms of g, t₁, and t₂.

(c) Express the horizontal distance d in terms of u, θ, t₁, and t₂.

6.  A particle starts from rest at x = 0 and moves along the x-axis. Its acceleration a varies with time t as shown by the relation a(t) = a_0( 1 – (t)/(T_0) ) for 0 ≤ t ≤ T₀, where a₀ = 6 m/s² and T₀ = 4 s.

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

(b) Find the maximum velocity attained by the particle during this interval.

(c) Calculate the total distance covered by the particle at t = T₀.

7.  A particle is projected horizontally with speed u = 20 m/s from the top of an inclined plane making an angle α = 30^(∘) with the horizontal. The incline falls away below the point of projection. Taking g = 10 m/s²:

(a) Derive the distance R along the inclined plane where the particle lands.

(b) Calculate the time of flight T before impact.

(c) Find the angle made by the velocity vector with the inclined plane just before impact.

8.  The motion of a body is described by the equation (dv)/(dt) = 6 – 3v, where v is in m/s and t is in seconds. At t = 0, v = 0 and x = 0.

(a) Find the terminal (maximum possible) velocity of the body.

(b) Derive v(t) as an explicit function of time.

(c) Find the displacement x(t) as a function of time.

9.  An elevator ascends with a uniform downward deceleration a = 2 m/s². A person inside drops a coin from a height of 1.8 m above the floor. Taking g = 10 m/s²:

(a) Determine the effective acceleration experienced by the coin in the frame of the elevator.

(b) Calculate the time taken for the coin to hit the floor.

(c) Find the displacement of the coin in the ground frame during this fall if the initial upward velocity of the elevator was 8 m/s.

10.  A particle moves in the x-y plane with acceleration a = (4tî+6ĵ) m/s². At t = 0, its position is r₀ = (2î−ĵ) m and its velocity is v₀ = (3î+2ĵ) m/s.

(a) Find the position vector r(t) at any time t.

(b) Find the speed of the particle at t = 2 s.

(c) Determine the equation of the trajectory in Cartesian coordinates y(x) by eliminating t (approximate or implicit form allowed).

11.  A stone is thrown from ground level such that its horizontal range is R = 80 m and maximum height reached is H = 20 m. Taking g = 10 m/s²:

(a) Find the angle of projection θ.

(b) Calculate the initial velocity u.

(c) Find the radius of curvature of the trajectory at the highest point.

12.  Two particles A and B are located at points (0,0) and (0,100 m) in the x-y plane at t = 0. Particle A moves with a constant velocity v_(A) = 10î m/s, while particle B moves with a constant velocity v_B = ( 5i – 5√(3)j )~m/s.

(a) Find the relative velocity v_(AB) = v_(A) − v_(B).

(b) Determine the distance of closest approach between A and B.

(c) Calculate the time t at which they are closest to each other.

13.  The velocity of a particle moving along the x-axis is given by v(x) = k√(x^2) + 4, where k = 2 s⁻¹ and x is in meters.

(a) Find the acceleration of the particle as a function of position x.

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

(c) Assuming x(0) = 0, find the position x(t) as a function of time.

14.  A boy throws a ball vertically upward inside a train moving horizontally with a constant acceleration a₀ = 3 m/s². The initial vertical velocity given to the ball relative to the train is u_(y) = 20 m/s. Taking g = 10 m/s²:

(a) Calculate the time of flight of the ball before it returns to the height of release.

(b) Determine how far from the boy’s hand the ball lands inside the train car.

(c) Describe the path of the ball as observed by (i) the boy inside the train and (ii) an observer standing on the ground.

15.  A point moves along a circle of radius R = 5 m with a constant tangential acceleration a_(t). If the velocity of the point is v = 10 m/s at the end of the second revolution after beginning of motion:

(a) Calculate the tangential acceleration a_(t).

(b) Find the centripetal acceleration a_(c) at the end of the second revolution.

(c) Find the total acceleration vector magnitude at that instant.

16.  A projectile is fired at speed u at an angle θ above the horizontal. At time t₁, its velocity vector makes an angle + 30^(∘) with the horizontal, and at time t₂, it makes an angle − 30^(∘).

(a) Express t₁ and t₂ in terms of u, θ, and g.

(b) Calculate the time interval Δt = t₂ − t₁.

(c) Show that during this interval Δt, the average velocity vector is purely horizontal.

17.  A car A moving at 30 m/s is pursuing car B moving at 20 m/s in the same direction on a straight road. When car A is 100 m behind car B, car A applies brakes, giving it a uniform deceleration of 2 m/s².

(a) Write the position equations for both cars in terms of time t.

(b) Determine whether car A will collide with car B.

(c) If they collide, find the time of collision; if not, find the distance of closest approach.

18.  The acceleration of a particle is defined by a = − 4x, where a is in m/s² and x is in meters. At t = 0, x = 2 m and v = 0.

(a) Derive an expression for velocity v as a function of position x.

(b) Find the maximum speed of the particle and the coordinates where it occurs.

(c) Identify the type of motion and determine its period T.

19.  A particle is launched with an initial speed u = 50 m/s at an angle θ = 53^(∘) above the horizontal (sin 53^(∘) = 0.8, cos 53^(∘) = 0.6). Taking g = 10 m/s²:

(a) Find the speed of the particle at t = 2 s.

(b) Calculate the angle of the velocity vector with the horizontal at t = 2 s.

(c) Find the time t when the velocity vector becomes perpendicular to the initial velocity vector v₀.

20.  A block is placed at rest on a long conveyor belt moving at a constant 4 m/s. The block accelerates uniformly at 2 m/s² until its speed matches the belt.

(a) Find the time taken for the block to reach the belt speed.

(b) Find the distance travelled by the block in that time.

(c) Find the distance travelled by the belt over the same interval, and hence the slip between them.

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.