L2 Calculus
Question Bank
A particle's displacement is $s(t) = t^3 - 6t^2 + 9t$. Find its velocity and acceleration at $t=2$.
The distance, $s$ metres, of a particle from a fixed point after $t$ seconds is given by $s(t) = 2t^3 - 5t^2 + 8t + 1$. Find expressions for its velocity and acceleration.
The velocity $v$ m s⁻¹ of an object $t$ seconds after it passes a fixed point can be modelled by the function
$v(t) = 3t^2 - 8t + 5$
Find the equation for the acceleration of the object.
The velocity $v$ m s⁻¹ of a car $t$ seconds after it starts moving is given by
$v(t) = 5t^2 + 2t + 1$
Find the acceleration of the car when $t = 3$ seconds.
The velocity $v$ m s⁻¹ of a train $t$ seconds after it passes a station is given by
$v(t) = 2t^3 - 15t^2 + 24t$
At what time is the acceleration of the train equal to zero?
The position $s$ metres of an object $t$ seconds after it starts moving is given by
$s(t) = t^3 - 6t^2 + 9t$
At what time(s) is the object stationary?
The velocity of a particle is given by $v(t) = 8t - 0.5t^2$ m/s. Find the acceleration of the particle when its velocity is 14 m/s.
The acceleration of an object is given by $a(t) = 6t - 2$ m/s². Its initial velocity is 5 m/s. What is its velocity after 4 seconds?
A remote-controlled car's distance from a wall is given by $s(t) = 12t - t^3$ for $0 \le t \le 3$. Find the car's maximum distance from the wall.
A boat's acceleration is $a(t) = 0.5$ km/h². When it starts fishing ($t=0$), its speed is 3 km/h moving away from port, and it is 80 km from port. Find its distance from the port after 6 hours.
An object is dropped (from rest) off a cliff with a constant acceleration down at a rate of 9.8m/s². Initially the object is 10 m from the ground. Find when the object hits the ground. You must use calculus to find your answer.
A train is travelling at 30 m/s when the driver applies the brakes, causing a constant deceleration of 2 m/s². How far does the train travel before it comes to a complete stop?
A particle moves along a straight line. Its displacement from a fixed point O after $t$ seconds is
$s(t) = 2t^3 - 21t^2 + 60t$
metres. Find the distance between the two positions where the particle is momentarily at rest.
A drone is flying away from its base. Its acceleration is $a(t) = 4t - 10$ m/s². Two seconds after leaving the base, its velocity is 8 m/s. How far is the drone from its base after 5 seconds?
A car is travelling at $25$ m s⁻¹ when the driver sees a fallen tree across the road, $60$ m ahead.
The driver takes $0.8$ seconds to react, and the car keeps the same speed during this time. The driver then brakes, and the car slows down with a constant deceleration of $6.25$ m s⁻².
Use calculus to decide whether the car stops before it reaches the tree.
An automated train travels between two stations. As it leaves Station 1, its velocity is modelled by the function $v(t) = 0.1t^2 + 0.5t$ m s⁻¹, where $t$ is the time in seconds after leaving the station.
At the exact moment the train's acceleration reaches 2.5 m s⁻², the train's braking system is scheduled to engage to bring it to a stop at Station 2. The braking provides a constant deceleration.
The entire journey from Station 1 to Station 2 takes a total of 50 seconds.
Calculate the constant deceleration required from the braking system to ensure the train stops exactly at Station 2.
A model rocket is launched vertically from the ground. Its motion occurs in two distinct stages.
- Stage 1 (First 10 seconds): The rocket engine burns, producing an acceleration of $a(t) = 0.6t$ m s⁻² for $0 \le t \le 10$, where $t$ is the time in seconds after launch. The rocket starts from rest at ground level.
- Stage 2 (After 10 seconds): At $t = 10$ seconds, the first-stage engine cuts out and a second, smaller engine fires, providing a constant acceleration of 4 m s⁻².
Calculate the total height of the rocket above the ground after it has been travelling for a total of 25 seconds.
A bus pulls away from a bus stop, starting from rest. Its acceleration is a constant $1$ m s⁻².
At the moment the bus starts, a cyclist riding at a steady $5$ m s⁻¹ is already $12$ m past the bus stop, travelling in the same direction.
Use calculus to find how far from the bus stop the bus overtakes the cyclist.
A radio-controlled car moves along a straight track. Its velocity $t$ seconds after it leaves the start line is
$v(t) = 0.3t^2 - 1.8t + 2.4$
metres per second. A positive velocity means the car is moving away from the start line.
Find the total distance the car travels in the first 5 seconds.