Kinematics of Motion in a Straight Line: Mathematics 9709 (Cambridge International AS & A Level)

A go-kart passes a marker post on a straight track with a velocity of $3\text{ m s}^{-1}$ and then accelerates uniformly at $1.5\text{ m s}^{-2}$.

Find the velocity of the go-kart $6$ seconds after it passes the marker post.

A cyclist rides along a straight road for $40$ seconds, starting from rest at a fixed point $O$. The velocity–time graph of her motion consists of three straight-line stages:

• Stage 1 ($0 \le t \le 10$): the velocity increases uniformly from $0$ to $8\text{ m s}^{-1}$.
• Stage 2 ($10 \le t \le 30$): the velocity stays constant at $8\text{ m s}^{-1}$.
• Stage 3 ($30 \le t \le 40$): the velocity decreases uniformly from $8\text{ m s}^{-1}$ back to $0$.

(a) Find the acceleration of the cyclist during Stage 1 and during Stage 3. [2]

(b) By considering the area under each stage of the velocity–time graph, find the total distance travelled by the cyclist during the $40$ seconds. [4]

(c) Find the average velocity of the cyclist over the whole $40$ seconds. [2]

A delivery robot moves along a straight corridor, starting from rest at a fixed point $O$. Its acceleration $a\text{ m s}^{-2}$, at time $t$ seconds after leaving $O$, is given by $a = 6 - 3t \quad \text{for } 0 \le t \le 4.$

(a) Show that the velocity of the robot at time $t$ is given by $v = 6t - 1.5t^2$. [3]

(b) Find the maximum velocity attained by the robot, and the value of $t$ at which it occurs. Justify why this gives a maximum rather than a minimum. [3]

(c) Find the total distance travelled by the robot during $0 \le t \le 4$. [2]

A stone is thrown vertically upward from ground level with an initial speed of $25\text{ m s}^{-1}$. The stone is modelled as a particle moving in a straight vertical line, and air resistance is ignored. Take $g = 10\text{ m s}^{-2}$, and take the upward direction as positive.

(a) Find the greatest height above the ground reached by the stone. [3]

(b) Find the total time taken for the stone to return to the ground. [3]

(c) Find the speed at which the stone hits the ground. [2]

A particle moves along a straight line. Its displacement $s$ metres from a fixed point $O$, $t$ seconds after it starts to move, has a displacement–time graph made of two straight-line stages:

• For $0 \le t \le 4$, $s$ increases uniformly from $0$ m to $20$ m.
• For $4 \le t \le 10$, $s$ decreases uniformly from $20$ m to $5$ m.

Find the velocity of the particle during the interval $4 \le t \le 10$.

A train travels along a straight, horizontal section of track at a constant velocity of $24\text{ m s}^{-1}$. As it approaches a station, the driver applies the brakes, giving the train a constant deceleration. The train comes to rest after travelling a further $180$ m.

(a) Find the deceleration of the train. [3]

(b) Find the time taken for the train to come to rest after the brakes are applied. [3]

(c) Find the average velocity of the train while it is decelerating, and use it to verify your answer to part (b). [2]

A skateboarder starts from rest at the top of a straight ramp and accelerates uniformly down the ramp at $0.4\text{ m s}^{-2}$.

Find the distance she has travelled after $5$ seconds.

A particle moves along a straight line. Its velocity–time graph consists of two straight-line stages:

• For $0 \le t \le 4$, the velocity decreases uniformly from $8\text{ m s}^{-1}$ to $0$.
• For $4 \le t \le 6$, the velocity continues to decrease uniformly from $0$ to $-6\text{ m s}^{-1}$ (the particle is now moving back towards its starting point).

Find the total displacement of the particle from its starting point at $t=6$.

A particle $P$ moves along a straight line so that its displacement $s$ metres from a fixed point $O$, at time $t$ seconds ($t\ge0$), is given by $s = t^3 - 6t^2 + 9t.$

(a) Find expressions for the velocity $v$ and the acceleration $a$ of $P$ at time $t$. [3]

(b) Find the values of $t$ at which $P$ is instantaneously at rest. [3]

(c) Find the acceleration of $P$ at each of the values of $t$ found in part (b), and use these to state, in each case, whether $P$ is about to change direction. [2]

Car $A$ starts from rest at a fixed point $O$ on a straight road and moves with constant acceleration $2\text{ m s}^{-2}$. Car $B$ travels along the same road in the same direction at a constant velocity of $20\text{ m s}^{-1}$, and passes through $O$ exactly $5$ seconds after Car $A$ sets off. Let $t$ be the time in seconds measured from the instant Car $A$ sets off.

(a) Write down expressions, in terms of $t$, for the displacement from $O$ of Car $A$, and of Car $B$ (valid for $t\ge5$). [3]

(b) Show that the times at which the two cars are at the same distance from $O$ satisfy $(t-10)^2=0$, and hence find this value of $t$. [3]

(c) State what a repeated root tells you about the motion of the two cars, and find the distance from $O$ at which the two cars are momentarily at the same position. [2]