3.8:
Kinematic Equations – II
The first kinematic equation expresses velocity as a function of the initial velocity and the change in velocity.
For deriving the second kinematic equation, two equations for average x velocity are used during an interval from t equal to zero to a later time t. The first equation for average x velocity is the change in displacement over the change in time, as discussed in previous lessons. Here, the initial position is denoted by x0 at time t equal to zero and x at later time t.
Assuming acceleration to be constant, the average x velocity can also be represented as the average of the velocities at the initial and final times. Substituting the first kinematic equation here, an equation for the position x as a function of time is obtained.
This equation shows that the position of an object at any time t is the sum of its initial position, the distance moved under constant initial velocity, and the distance traveled during the change in velocity.
3.8:
Kinematic Equations – II
The second kinematic equation expresses the final position of an object in terms of its initial position, the distance traveled with the initial constant velocity, and the distance traveled due to a change in velocity. Similar to the first kinematic equation, this equation is also only valid when the acceleration is constant throughout the motion of an object.
Suppose a car merges into freeway traffic on a 200 m long ramp. If its initial velocity is 10 m/s and it accelerates at 2 m/s2, then the time taken by the car to travel the 200 m long ramp can be calculated using the second kinematic equation. Here, the known quantities are the distance of 200 m, the initial velocity of the car of 10 m/s, and the acceleration of 2 m/s2. Using the second kinematic equation
and substituting the known quantities in the above equation, we get
Using the quadratic formula to solve for time yields two solutions: t = 10 s and t = −20 s. A negative value for time is unreasonable, since that would mean the event happened 20 s before the motion began, therefore we can discard that solution. Thus, the time taken by the car to travel the 200 m ramp is 10 s.
This text is adapted from Openstax, University Physics Volume 1, Section 3.4: Motion with Constant Acceleration.