Find the initial speed of the ball.

Find the angle of elevation of the ball as it is launched.

Find the time taken for the snail to reach the point $(10, 2)$.

Given that particles A and B start at the same point, find the displacement function $s_{\text{B}}$ for particle B.

Find the value of $t$ when particle A first changes direction.

Find the total distance travelled by particle A in the first 3 seconds.

Write down the first value of $t$ at which P changes direction.

Find the magnitude of the particle’s acceleration at $6$ seconds.

Show that the acceleration vector of $P$ is never parallel to the position vector of $P$.

Hence show that the snail reaches the point $(10, 2)$ before the water does.

Juri starts at a height of $60$ metres and finishes at $\text{F}$, where $x=f$.

Sketch a possible diagram of the hill on the following pair of coordinate axes.

Find the velocity vector of $P$.

When $\text{W}$ has an $x$-coordinate equal to $1$, find the horizontal component of the velocity of $\text{W}$.

Determine the first time t₁ at which P has zero velocity.

Calculate the vertical distance Xavier travelled in the first 10 seconds.

Find the total distance travelled in the first 5 seconds.

Find the total distance travelled by P.

Find the initial velocity of P.

Determine the value of $h$.

Write down the values of $t$ when $a=0$.

Find the maximum speed of P.

Find the value of the acceleration of P at time t₁.

Find the total distance travelled by P, for $0⩽t⩽8$.

Note:     In this question, distance is in metres and time is in seconds.

A particle moves along a horizontal line starting at a fixed point A. The velocity $v$ of the particle, at time $t$, is given by $v(t)=\frac{2t^2-4t}{t^2-2t+2}$, for $0⩽t⩽5$. The following diagram shows the graph of $v$

There are $t$-intercepts at $(0,0)$ and $(2,0)$.

Find the maximum distance of the particle from A during the time $0⩽t⩽5$ and justify your answer.

Show that the magnitude of the velocity of the ice-skater at time $t$ is given by

$a {\text{e}}^{bt}\sqrt{\left(1+b^2\right)}$.

Find the value of $p$.

Find the value of $t$ when particle A first reaches point P.

Find when the particle is at rest.

Hence or otherwise, find all possible values of $t$ for which the velocity of P is decreasing.

A second particle Q also moves along a straight line. Its velocity, $v_{\text{Q}}\text{ m}{\text{s}}^{-1}$ after $t$ seconds is given by $v_{\text{Q}}=\sqrt{t}$ for $0⩽t⩽8$. After $k$ seconds Q has travelled the same total distance as P.

Find $k$.

Find the initial displacement of particle A from point P.

Find the greatest speed of $\text{P}$ in the interval $0\le t\le 6$.

Find the total distance travelled by the particle.

Find an expression for the velocity of P at time $t$.

Write down the number of times that the acceleration of P is 0 m s⁻² .

(i)     Find the value of $q$.

(ii)     Hence, find the speed of P when $t=q$.

A particle moves along a horizontal line such that at time $t$ seconds, $t$ ≥ 0, its acceleration $a$ is given by $a$ = 2$t$ − 1. When $t$ = 6 , its displacement $s$ from a fixed origin O is 18.25 m. When $t$ = 15, its displacement from O is 922.75 m. Find an expression for $s$ in terms of $t$.

How many complete revolutions has the ball completed from $t=0$ to the instant at which the string breaks?

Find the total distance travelled by P when its velocity is increasing.

Find $t_1$ and $t_2$.

Find the initial velocity of $P$.

The particle starts from the origin $\text{O}$. Find an expression for the displacement of $\text{P}$ from $\text{O}$ at time $t$ seconds.

(i)     Find the total distance travelled by P between $t=1$ and $t=p$.

(ii)     Hence or otherwise, find the displacement of P from A when $t=p$.

Find the magnitude of the velocity of the ice-skater when $t=2$.

At a point $\text{P}$, the ice-skater is skating parallel to the $y$-axis for the first time.

Find $\text{OP}$.

Find the value of $a$ and the value of $b$.

Find the value of $s\left(4\right)-s\left(2\right)$.

A particle moves in a straight line such that at time $t$ seconds $(t⩾0)$, its velocity $v$, in $\text{m}{\text{s}}^{-1}$, is given by $v=10t{\text{e}}^{-2t}$. Find the exact distance travelled by the particle in the first half-second.

Find his velocity when $t=15$.

Find an expression for the acceleration of P at time t.

Find the other value of $t$ when particles A and B meet.

Find the times when $\text{P}$ is at instantaneous rest.

Find the total distance travelled by $\text{P}$ in the interval $0\le t\le 4$.

Find the displacement of the particle when $t=t_1$

Find the acceleration of the particle when $t=2$.

Find an expression for the velocity of the ball at time $t$.

Find the velocity vector at time $t$.

Hence show that the velocity of the ball is always perpendicular to the position vector of the ball.

Find an expression for the acceleration of the ball at time $t$.

Show that the ball is moving in a circle with its centre at $\text{O}$ and state the radius of the circle.

Find the value of $t$ at the instant the string breaks.

Identify the $x$ value of the point where $|h\prime (x\left)\right|$ has its maximum value.

Interpret this point in the given context.