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In this lesson, we will learn how to calculate instantaneous acceleration as the rate of velocity change in a time interval that approaches zero.

Q1:

A cheetah can accelerate from rest to a speed of 30.0 m/s in 7.00 s. What is its acceleration?

Q2:

A bullet in a gun is accelerated from the firing chamber to the end of the barrel at an average rate of 6 . 2 0 × 1 0 5 m/s^{2} for a time of 8 . 1 0 × 1 0 − 4 s. What is the speed of the bullet when it leaves the gun’s barrel?

Q3:

A volume of blood is accelerated by the heart’s pumping action and moves 1.80 cm in a straight line along an artery. The blood accelerates from rest to a speed of 30.0 cm/s as it moves through the artery. How much time is the blood accelerated for?

Q4:

An aeroplane, starting from rest, moves down the runway at constant acceleration for 27 s and then takes off at a speed of 85 m/s. What is the average acceleration of the plane along the runway?

Q5:

A sprinter runs a 100-m race in 9.74 s. The sprinter accelerated for 3.22 s to reach her maximum speed, which she maintained for the rest of the race.

Find the sprinter’s acceleration.

Find the sprinter’s maximum speed.

Q6:

A commuter backs her car out of her garage with an acceleration of 1.21 m/s^{2}.

How long does it take her to reach a speed of 3.30 m/s?

From a speed of 3.30 m/s, the commuter brakes to a stop in 0.550 s. What is her acceleration?

Q7:

The position of a particle along the 𝑥 -axis varies with time according to the equation 𝑥 ( 𝑡 ) = 1 . 5 − 3 . 3 𝑡 2 m .

What is the velocity of the particle at 𝑡 = 2 . 7 s ?

What is the velocity of the particle at 𝑡 = 4 . 3 s ?

What is the acceleration of the particle at 𝑡 = 2 . 7 s ?

What is the acceleration of the particle at 𝑡 = 4 . 3 s ?

Q8:

The velocity of a particle moving along the 𝑥 -axis varies with time according to 𝑣 ( 𝑡 ) = 𝐴 + 𝐵 𝑡 − 1 , where 𝐴 = 1 . 3 0 / m s , 𝐵 = 0 . 3 1 m , and 2 . 0 ≤ 𝑡 ≤ 9 . 0 s. At 𝑡 = 2 . 0 s , 𝑥 = 0 . 0 0 m .

Determine the acceleration of the particle at 𝑡 = 3 . 5 s .

Determine the acceleration of the particle at 𝑡 = 6 . 7 s .

Determine the displacement from 𝑥 = 0 . 0 0 m of the particle at 𝑡 = 3 . 5 s .

Determine the displacement from 𝑥 = 0 . 0 0 m of the particle at 𝑡 = 6 . 7 s .

Q9:

A motorcycle accelerates in a straight line from rest to a speed of 26.8 m/s in 3.90 s.

What is the average acceleration of the motorcycle?

How far does the motorcycle travel while accelerating?

Q10:

A light-rail commuter train traveling in a straight line accelerates at a rate of 1.93 m/s^{2}.

How much time is required for the train to reach a speed of 74.0 km/h?

If the train is traveling at 74.0 km/h and decelerates to rest at a rate of 1.18 m/s^{2}, how much time passes before the train stops?

The train can decelerate more rapidly in emergencies, coming to rest from 74.0 km/h in 11.3 s. What is the emergency acceleration rate?

Q11:

A racehorse passing through a starting gate accelerates from rest to a velocity of 15.0 m/s due west in a time interval of 1.80 s. Find the racehorse’s average acceleration. Assume that east corresponds to positive displacement.

Q12:

A particle that is in motion has a changing velocity that is modeled by the function 𝑣 ( 𝑡 ) = 2 0 𝑡 − 5 . 0 𝑡 2 m/s.

Find the instantaneous velocity of the particle at 𝑡 = 1 . 0 s .

Find the instantaneous velocity of the particle at 𝑡 = 2 . 0 s .

Find the instantaneous velocity of the particle at 𝑡 = 3 . 0 s .

Find the instantaneous velocity of the particle at 𝑡 = 5 . 0 s .

Find the instantaneous acceleration of the particle at 𝑡 = 1 . 0 s .

Find the instantaneous acceleration of the particle at 𝑡 = 2 . 0 s .

Find the instantaneous acceleration of the particle at 𝑡 = 3 . 0 s .

Find the instantaneous acceleration of the particle at 𝑡 = 5 . 0 s .

Q13:

A particle that is at rest at 𝑡 = 0 s accelerates according to the function 𝑎 ( 𝑡 ) = 5 − 1 0 𝑡 m/s^{2}. At what nonzero value of 𝑡 is the particle at rest?

Q14:

Protons in a linear accelerator are accelerated from rest to a speed of 2 . 0 × 1 0 − 7 m/s in a time of 1 . 0 × 1 0 − 4 s. What is the magnitude of the average acceleration of the protons?

Q15:

A car merges into freeway traffic on a 200 m long ramp. If its initial velocity is 10.0 m/s and it accelerates at 2.00 m/s^{2}, how long does it take the car to travel the 200 m up the ramp?

Q16:

On dry concrete, a car can decelerate at a rate of 7.00 m/s^{2}, whereas on wet concrete it can decelerate at only 5.00 m/s^{2}. The car has an initial velocity of 30.0 m/s. The car’s stopping distance is the distance that it travels before coming to rest from its initial velocity. The stopping distance is affected by the driver’s reaction time, which is the time interval between the driver making the decision to apply the car’s brakes and moving her foot to apply them. The driver’s reaction time is 0.500 s.

What is the car’s stopping distance on dry concrete, neglecting the driver’s reaction time?

What is the car’s stopping distance on wet concrete, neglecting the driver’s reaction time?

What is the car’s stopping distance on dry concrete?

What is the car’s stopping distance on wet concrete?

Q17:

What quantity is acceleration the first derivative of?

Q18:

A spaceship has left Earth’s orbit and is on its way to the Moon. It accelerates at 20.00 m/s^{2} for 120.0 s and moves a distance of 1 . 0 0 0 × 1 0 m in that time interval.

What is the speed of the spaceship before its acceleration?

What is the speed of the spaceship after its acceleration?

Q19:

A cart is constrained to move along a straight line. A varying net force along the direction of motion is exerted on the cart. The cart’s velocity 𝑣 as a function of time 𝑡 is shown in the graph. The five labeled points divide the graph into four sections. Which of the following correctly ranks the magnitude of the average acceleration of the cart during the four sections of the graph?

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