Velocity in accelerated motion

The equation defining acceleration can be rewritten as a model that describes how the velocity changes over time. The result is equation (4.2), which gives the instantaneous velocity v at time t under the assumption that the acceleration a is constant. Read the text aloud Show Where are<br/><i>t<sub>i</sub></i> and <i>t<sub>f</sub></i>?
(4.2) v= v 0 +at
v  = velocity (m/s)
v0  = initial velocity (m/s)
a  = acceleration (m/s2)
t  = time (s)
Velocity
for constant acceleration
In this equation we set Δt = t by assuming the motion starts at t = 0. The initial velocity v0 is the velocity when t = 0. Read the text aloud Show Comparing a velocity model to an equation for a straight line
A cart traveling at 1 m/s reaches a hill and accelerates down the hill at 0.5 m/s2. What is the velocity of the cart 3 s after it starts accelerating? Read the text aloud
What is the velocity of the cart after three seconds?
Asked: instantaneous velocity v
Given: initial velocity of v0 = 1 m/s, acceleration a = 0.5 m/s2, and time t = 3.0 s
Relationships: v = v0 + at
Solution: = 1 m/s + (0.5 m/s2)(3.0 s)
  =  2.5 m/s
In the above example the velocity and acceleration are in the same direction—both are positive. The speed increases from 1 to 2.5 m/s. Acceleration can also decrease an object’s speed when the sign of the velocity is different from the sign of the acceleration. For example, an acceleration of −1 m/s2 adds −1 m/s to the velocity each second. This would decrease a positive velocity. It would also make a negative velocity more negative—that is, faster in the negative direction. Read the text aloud
A cart traveling at 2 m/s along a level surface reaches an upward sloping hill and accelerates at −0.5 m/s2. What is the velocity of the cart 3 s after it starts climbing the hill? Read the text aloud
What is the velocity of the cart after three seconds?
Asked: instantaneous velocity v
Given: initial velocity of v0 = 2 m/s, acceleration of a = −0.5 m/s2, and time of t = 3 s
Relationships: v = v0 + at
Solution: v  = 2 m/s + (−0.5 m/s2)(3.0 s)
   =  0.5 m/s
The model of motion given by equation (4.2) applies to the instantaneous velocity at time t. Translated to an English sentence the equation tells us that the velocity v at time t is the initial velocity v0 plus the change in velocity due to acceleration a applied every second for t seconds. The assumption of constant acceleration means that the change in velocity each second is the same. Read the text aloud
Can you describe a situation in which an object’s acceleration is negative but its speed is increasing? Show

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