kinematics in physics

Question

An object begins moving at x= 0 from rest in the positive x direction with constant acceleration. It travels D meters in T seconds. Its final speed is V meters per second.

(a) What is the instantaneous speed of the object at time t= 1/2 T? Express your answer in terms of V.

(b) What is the position x of the object at time t=1/2 T? Express your answer in terms of D.

(c) What is the instantaneous velocity of the object when its position is x=D/2? Express your answer in terms of V.

Answer 1

**The easiest way to do these problems is to look at the relationship between the two variables in the equation. These variables are related directly (a), by a power of 2 (b) and by a square root (c). Below I worked the problems out more explicitly.**

a) We are looking for the new final velocity halfway through the time. Use the kinematic equation V = v₀ + aT. Here v₀ = 0, so v_f = aT. It should be clear then that if we plug in 0.5T for T, we will get 0.5aT, which is 1/2 of the original V, so the answer would be V/2 or 0.5V.

b) This time we are looking for the position halfway through the time. The kinematic equation to look at this time is D = D₀ + v₀T + 0.5aT². Since D₀ and v₀ are both 0, this simplifies to D = 0.5aT². If we plug in 0.5T for T, we will get D = (0.25)(0.5aT²) which is 1/4 of the original D, so the answer would be D/4 or 0.25D.

c) Now we are looking for the final velocity halfway through the displacement. Use the kinematic equation V² = v₀² + 2a(D - D₀). Since D₀ and v₀ are both 0, this simplifies to V² = 2aD or V = √(2aD). If we plug in 0.5D for D, we will get V = √((0.5)(2aD)) which is (√2/2)(√(2aD)) or (√2/2)V which is about 0.71V.