Concept #1: Motion in 2D

Practice: An object moves up with a constant 8 m/s while moving to the left with a constant 6 m/s. Calculate the magnitude and direction of the object’s (total) velocity.

Practice: An object is launched with an initial velocity of 50 m/s directed 37 degrees above the horizontal. Calculate the X and Y components of the object’s initial velocity.

Practice: A car is initially at rest at (0, 0) meters on an X-Y plane. The car then accelerates for 4 s with a constant 6 m/s^{2} directed in the +x-axis. The car then accelerates for 5 s with a constant 5 m/s^{2} directed at 53° above the +x-axis. Find the magnitude and direction of the car’s velocity after the two accelerations (after 8 s → hint: use v = vo + at).

Superman's arch nemesis Lex Luthor pushes Lois Lane horizontally with a speed of 5 m/s off a 35 m tall building. Superman stands on the ground just below the edge of the rood. Ignore air resistance.
What will be the vertical component of Lois' velocity just before Superman catches her?

A hockey puck slides horizontally off the edge of a table with an initial velocity of 20.0 m/s. The height of the table above the ground is 2.00 m. What is the magnitude of the vertical component of the velocity of the puck just before it hits the ground?
[A] 6.26 m/s
[B] 12.5 m/s
[C] 16.3 m/s
[D] 19.6 m/s
[E] 22.4 m/s

The velocity vector v(t) measured in m/s for a particle at time t is given as v(t) = [(3t + 4t3)î − 2ĵ] m/s, where t is in seconds.
[a] Find the units of the coefficients 3, 4, and 2 in the vector v.
[b] Find the average acceleration in the interval t = 0 to t = 2 seconds.
[c] Find the acceleration at t = 2 s.
[d] Say that at t = 0 the particle is at the origin, r(0) = 0. Find the position vector r(t).
[e] Eliminate t to give the equation of the trajectory x as a function of y.

Using a reference frame with the origin at the take-off airport, the positive x-axis due East, and the positive y-axis due North, the acceleration a of an airplane as a function of time can be described as:
a = (αt)î + (βt 4 − γt)ĵ,
with α, β, and γ positive and constant.
Assuming that the airplane takes off from the airport at time t = 0 with zero initial velocity:
a) What are the units of α, β, and γ?
b) Find the time(s) when the airplane position is directly NE of the airport.
c) Find the trajectory of the plane, y(x).
Write your results in terms of α, β, and γ. Remember to check the dimensions/units for each answer.

The velocity of an airplane as a function of time can be written as v(t) = bi − 3ct2j where b and c are positive constants. Provide your answers in terms of b, c, and t when necessary.
[a] What are the SI units of b and c?
[b] Find an expression for the position r(t) of the airplane as a function of time assuming that the airplane is at the origin at t = 0.
[c] Find an expression for the acceleration a(t) of the airplane as a function of time
[d] Find the trajectory of the airplane (y vs. x).

A motorcycle rider wants to try the following stunt: she will start from rest and accelerate her bike at constant acceleration a0 on a horizontal platform of length L; she will then jump with her bike trying to land on a second platform placed a distance y0 below the first one.
[a] What is the speed of the rider at the end of the first platform?
[b] What is the maximum distance at which the second platform can be placed for the rider to reach it?
[c] What is the speed of the rider when she reaches the second platform?

Two balls roll off a 1.1 m high table with horizontal velocities of 2.5 m/s and 5.0 m/s, respectively. they reach the edge of the table at the same time. Which ball hits the ground first?
A) The ball that travels the shorter distance hits the ground first.
B) The slower ball hits the ground first.
C) The faster ball hits the ground first.
D) Both balls hit the ground at the same time.

Two balls roll off a 1.1 m high table with horizontal velocities of 2.5 m/s and 5.0 m/s, respectively. they reach the edge of the table at the same time. What is the horizontal velocity of the faster ball when it hits the ground?
A) 5.00 m/s
B) 1.10 m/s
C) 2.50 m/s
D) 4.64 m/s
E) 9.80 m/s

When fighting forest fires, airplanes work in support of ground crews by dropping water on the fires. A pilot is practicing by dropping a canister of red dye, hoping to hit a target on the ground below. If the plane is flying in a horizontal path 29.0 m above the ground and with a speed of 19.0 m/s, at what horizontal distance from the target should the pilot release the canister? Ignore air resistance.

Two balls roll off a 0.8 m high table with horizontal velocities of 2.0 m/s and 4.0 m/s, respectively. They reach the edge of the table at the same time. Which ball hits the ground first?A) Both balls hit the ground at the same time.B) The slower ball hits the ground first.C) The ball that travels the shorter distance hits the ground first.D) The faster ball hits the ground first.

Two balls roll off a 0.8 m high table with horizontal velocities of 2.0 m/s and 4.0 m/s, respectively. What is the horizontal velocity of the faster ball when it hits the ground?A) 9.80 m/sB) 3.96 m/sC) 0.80 m/sD) 2.00 m/sE) 4.00 m/s

When fighting forest fires, airplanes work in support of ground crews by dropping water on the fires. A pilot is practicing by dropping water on the fires. A pilot is practicing by dropping a canister of red dye, hoping to hit a target on the ground below. If the plane is flying in a horizontal path of 41.0 m above the ground and with a speed of 18.0 m/s, at what horizontal distance from the target should the pilot release the canister? Ignore air resistance.

A stuntman for a 007 movie scene needs to jump into the lake below while avoiding the ledge.1) If his initial velocity is horizontal, what is the time for the stuntman to fall 9.0m?2) What must be the minimum speed vo to pull off such a stunt?3) The stuntman knew that he can only run as fast as 1.2 m/s, and decided to perform this stunt by jumping with an upward angle of 45 degrees. What will be his landing point, and will that be greater than 1.75 meters?

A bullet is fired horizontally, and at the same instant a second bullet is dropped from the same height. Ignore air resistance. Compare the times of the fall of the two bullets.A) They hit at the same time.B) The fired bullet hits first.C) The dropped bullet hits first.D) Cannot tell without knowing the masses.