Acceleration, Velocity, Displacement, and Time

Learning Intentions

  • Learn how to calculate displacement, velocity, and time traveled under constant acceleration

Notes

  1. It is often useful to find the displacement, initial velocity, or final velocity without needing to calculate the time. Unfortunately, our equation for displacement includes time as a variable.
    \vec{d}\ =\ \vec{v}_0t\ +\ \frac{1}{2}\vec{a}t^2
    Luckily, we have an equation for acceleration that allows us to eliminate t as a variable, and add v1 as a variable.
    \vec{v}_1\ =\ \vec{v}_0\ +\ \vec{a}t
    Derive an equation that relates v0, v1, a, and d, without including t.







  2. Sometimes, we know the final velocity, the acceleration, and the time, and we want to find the displacement. How can the original displacement formula be modified to eliminate v0 as a variable, and add v1 as a variable?








Questions

  1. A juggler drops a ball from a height of 1.5 m above the ground.
    1. What will be the velocity of the ball just before it hits the ground?
    2. How long will it take to hit the ground?
    3. If sound travels through the air at 343 m/s, how long will it take for the sound of impact to reach the juggler's ears? Their ears are at a height of 1.8 m from the ground.
  2. A car accelerates at a constant rate from 15 m/s [E] to 35 m/s [E] in 12 seconds.
    1. What is the rate of acceleration?
    2. What distance does the car travel in this time?
  3. At the scene of an accident, there are skid marks 52 m [N]. It is known that the car came to a complete stop at the end of the skid marks, and the car's brakes can produce a deceleration of 9.1 m/s2. How fast was the car going when it started to brake?
  4. On an asteroid, an astronaut drops a hammer from a height of 1.5 m. If it takes 5.2 seconds for the hammer to hit the ground, what is the value of g for this asteroid?
  5. A car is going 21 m/s [W]. The driver slams on the brakes, decelerating at 8.3 m/s2.
    1. How fast is the car going after 3.0 seconds?
    2. How far does the car travel during these 3.0 seconds?


Answers


    1. v1 = -5.4 m/s = 5.4 m/s [down] = 20 km/h [down]
    2. t = 0.55 s
    3. t = 5.2 x 10-3 s = 5.2 milliseconds
  2. a
    1. a = 1.7 m/s2 [E]
    2. d = 300 m [E]
  3. v0 = 31 m/s [N] = 110 km/h [N]
  4. a = 0.11 m/s2 [down]

    1. v1 = 0 m/s
    2. d = 26 m [W]