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Subsection 3.1.2 Exercises

Exercises — Stage 1

1

Suppose you throw a ball straight up in the air, and its height from \(t=0\) to \(t=4\) is given by \(h(t)=-4.9t^2+19.6t\text{.}\) True or false: at time \(t=2\text{,}\) the acceleration of the ball is 0.

2

Suppose an object is moving with a constant acceleration. It takes ten seconds to accelerate from \(1\;\frac{\mathrm{m}}{\mathrm{s}}\) to \(2\;\frac{\mathrm{m}}{\mathrm{s}}\text{.}\) How long does it take to accelerate from \(2\;\frac{\mathrm{m}}{\mathrm{s}}\) to \(3\;\frac{\mathrm{m}}{\mathrm{s}}\text{?}\) How long does it take to accelerate from \(3\;\frac{\mathrm{m}}{\mathrm{s}}\) to \(13\;\frac{\mathrm{m}}{\mathrm{s}}\text{?}\)

3

Let \(s(t)\) be the position of a particle at time \(t\text{.}\) True or false: if \(s''(a) \gt 0\) for some \(a\text{,}\) then the particle's speed is increasing when \(t=a\text{.}\)

4

Let \(s(t)\) be the position of a particle at time \(t\text{.}\) True or false: if \(s'(a) \gt 0\) and \(s''(a) \gt 0\) for some \(a\text{,}\) then the particle's speed is increasing when \(t=a\text{.}\)

Exercises — Stage 2

For this section, we will ask you a number of questions that have to do with objects falling on Earth. Unless otherwise stated, you should assume that an object falling through the air has an acceleration due to gravity of 9.8 meters per second per second.

5

A flower pot rolls out of a window 10m above the ground. How fast is it falling just as it smacks into the ground?

6

You want to know how deep a well is, so you drop a stone down and count the seconds until you hear it hit bottom.

  1. If the stone took \(x\) seconds to hit bottom, how deep is the well?
  2. Suppose you think you dropped the stone down the well, but actually you tossed it down, so instead of an initial velocity of 0 metres per second, you accidentally imparted an initial speed of \(1\) metres per second. What is the actual depth of the well, if the stone fell for \(x\) seconds?
7

You toss a key to your friend, standing two metres away. The keys initially move towards your friend at 2 metres per second, but slow at a rate of 0.25 metres per second per second. How much time does your friend have to react to catch the keys? That is--how long are the keys flying before they reach your friend?

8

A car is driving at 100 kph, and it brakes with a deceleration of \(50000 \frac{\mathrm{km}}{\mathrm{hr}^2}\text{.}\) How long does the car take to come to a complete stop?

9

You are driving at 120 kph, and need to stop in 100 metres. How much deceleration do your brakes need to provide? You may assume the brakes cause a constant deceleration.

10

You are driving at 100 kph, and apply the brakes steadily, so that your car decelerates at a constant rate and comes to a stop in exactly 7 seconds. What was your speed one second before you stopped?

12

A pitching machine has a dial to adjust the speed of the pitch. You rotate it so that it pitches the ball straight up in the air. How fast should the ball exit the machine, in order to stay in the air exactly 10 seconds?

You may assume that the ball exits from ground level, and is acted on only by gravity, which causes a constant deceleration of 9.8 metres per second.

13

A peregrine falcon can dive at a speed of 325 kph. If you were to drop a stone, how high up would you have to be so that the stone reached the same speed in its fall?

14

You shoot a cannon ball into the air with initial velocity \(v_0\text{,}\) and then gravity brings it back down (neglecting all other forces). If the cannon ball made it to a height of 100m, what was \(v_0\text{?}\)

15

Suppose you are driving at 120 kph, and you start to brake at a deceleration of \(50 000\) kph per hour. For three seconds you steadily increase your deceleration to \(60 000\) kph per hour. (That is, for three seconds, the rate of change of your deceleration is constant.) How fast are you driving at the end of those three seconds?

Exercises — Stage 3

16

You jump up from the side of a trampoline with an initial upward velocity of \(1\) metre per second. While you are in the air, your deceleration is a constant \(9.8\) metres per second per second due to gravity. Once you hit the trampoline, as you fall your speed decreases by \(4.9\) metres per second per second. How many seconds pass between the peak of your jump and the lowest part of your fall on the trampoline?

17

Suppose an object is moving so that its velocity doubles every second. Give an expression for the acceleration of the object.

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