Warm-up: Position, Velocity, Acceleration

1. Consider an object that is held above the ground and dropped. It starts with an initial velocity [latex]v_0[/latex] (in [latex]m/s[/latex]). The object is also in free-fall, meaning it has a constant acceleration due to gravity (in [latex]m/{s}^2[/latex]). Write down the equation for velocity after a given time (in [latex]s[/latex]) as a function of initial velocity, time, and acceleration.

[latex]v = v_0 + a(\underline{\hspace{0.5cm}})[/latex]

2. Now consider an object that begins at an initial displacement [latex]d_0[/latex] (in [latex]m[/latex]) with a constant velocity [latex]v[/latex] (in [latex]m/s[/latex]). Write down the equation for displacement as a function of both time and velocity.

[latex]d = d_0 + v(\underline{\hspace{0.5cm}})[/latex]

3. Finally, consider an object that begins at [latex]d_0 = 0[/latex] [latex]m[/latex] with a velocity [latex]v_0 = 0[/latex] [latex]m/s[/latex] and has a constant acceleration. You observe the object’s movement for 4 seconds. Will the object cover more distance between [latex]t = 0[/latex] [latex]s[/latex] and [latex]t = 2[/latex] [latex]s[/latex] or in the interval from [latex]t = 2[/latex] [latex]s[/latex] and [latex]t = 4[/latex] [latex]s[/latex]. Why?

Try these practice questions below if you want to apply the equations above to a real world example. 

1. Consider an object that starts with a velocity [latex]v_0 = 0[/latex] [latex]m/s[/latex] and in free–fall, meaning its acceleration is given by [latex]a = 9.8 \text{ m/s}^2[/latex] in the downwards direction. Using the your formula from Question 1 of Kinematics: Warm Up, find the velocities after 1s and 4s (to one decimal place).

2. Now, consider an object begins at [latex]d_0 = 0[/latex] [latex]m[/latex] with a constant velocity [latex]v_0 = 2[/latex] [latex]m/s[/latex]. Using the values given in Question 2 of Kinematics: Warm Up, find the final displacement after 1s and 6s (to one decimal place).