DP Mathematics: Applications and Interpretation Questionbank

Find the velocity vector at time $t$.

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

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

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

Find $\text{OP}$.

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

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

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

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

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

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.

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

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

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

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

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.

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

Find the velocity vector of $P$.

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

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

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

Find the initial speed of the ball.

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

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.

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

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

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

Find $t_1$ and $t_2$.

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

Find his velocity when $t=15$.

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

Determine the value of $h$.

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 $t$.

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

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

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

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

Find the initial velocity of P.

Find the maximum speed of P.

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

Find the acceleration of P when it changes direction.

Find the total distance travelled by P.

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.

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

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

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 when the particle is at rest.

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

Find the total distance travelled by the particle.

Find the initial velocity of $P$.

Find the value of $p$.

(i)     Find the value of $q$.

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

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

Find the total distance travelled in the first 5 seconds.

Find the initial displacement of particle A from point P.

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

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

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

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

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