John starts running from $x = 0\ m$. His speed as a function of time $t$ is given by $v(t) = \frac{1}{20} \ t \ (16 − t)\ m/s$. For which of the below time instants is John’s speed a maximum?

Which position-time graph (in 1D) shows an average speed of $5\ m/s$ over the time interval when the object is in motion?

Lilly’s position vector $\overrightarrow{r}$ is a function of time $t$ given by the relation,

$\overrightarrow{r} (t)=2t \skew{2.5}\hat{i} + (5-t^2 ) \skew{4}\hat{j} \, \, m$

What is Lilly’s average velocity as she moves from $t = 2 \ s$ to $t = 4 \ s$?

An object of mass $2 \ kg$, starting from rest, accelerates uniformly for $5 \ s$. If the object covered a distance of $40 \ m$ during this time interval, what distance would it cover if its acceleration was 4 times larger?

The graph of the velocity $v$, as a function of time $t$, is shown below.

Which option would show the correct relation for the acceleration $a(t)$?

(Hint: The graph is a parabola of the form $y = b(x-a)^2$)

A projectile is launched at an angle of 20° with a speed of $10\ m/s$. Another projectile is launched at an angle of 40° with the same speed. The difference between the range of the two projectiles is,

(Use $g=10\ m/s^2$)

The velocity of particle $A$ is given by:

$\overrightarrow{v}_A (t)=(t-2) \skew{2.5}\hat{i} +5 \skew{4}\hat{j}$

The velocity of particle $B$ is given by:

$\overrightarrow{v}_B (t)=6 \skew{2.5}\hat{i} + (4-t) \skew{4}\hat{j}$

The velocity of $B$, as seen by $A$, is given by:

Which of the below statements is TRUE for the position-time graphs of two particles, $A$ and $B$?

The velocity $v(t)$ of a particle when passing through a medium is given by the formula,

$v(t) = \dfrac{mg}{γ} (1-e^{−\frac{γ \ t}{m}} )$

Where $g$ is the acceleration due to gravity, $γ$ is a constant, $m$ is the mass of the particle, and $t$ is the time. What is the maximum velocity that the particle will reach when passing through the medium?

A particle moving in 1D has velocity $v(t) = 6t + 3t^2 \ m/s$. What is the magnitude of displacement $\Delta r$ of the particle as it moves from $t = 1 \ s$ to $t=4 \ s$?

Questions 11 and 12 are based on the below information:

A projectile is launched, as shown in the diagram below.

What will be the velocity of the projectile after $t = 2 \ s$?

(Use $g = 10 \ m/s^2 $)

A projectile is launched, as shown in the diagram below.

What is the displacement of the projectile after $t = 4 \ s$?

A block starts decelerating at $3 \ m/s^2$ from time $ t = 0 \ s$, and covers $30 \ m$ before it stops. Which option is the closest to the average speed of the block?

An object is released from rest inside a medium, as shown below.

It experiences a downward acceleration $a(t) = 6at$ while moving through the medium, where $a$ is a constant. Find the time interval $\Delta t$ required for the object to fall through height $h$ in the medium.

Two boats, $I$ and $II$, are moving in a river as follows:

$I: \overrightarrow{r}(t)=8t \skew{2.5}\hat{i} +(7-2t) \skew{4}\hat{j} $

$II: \overrightarrow{r}(t)=4t^2 \skew{2.5}\hat{i} +2(t-t^2 ) \skew{4}\hat{j} $

Find the time instant $t$ when the two boats are traveling with the same velocity.