Let's go through the questions one by one from the image.

1) Kinematics in 1D

The first problem involves a car's motion along a straight line, given by the position function: $x(t) = 5t^2 + 2t + 1$ where $x$ is in meters and $t$ is in seconds.

• (a) Position at $t = 3$: Substitute $t = 3$ into the position function to find the car's position at that time.

• (b) Velocity at $t = 3$: The velocity is the first derivative of the position function, $v(t) = \frac{dx(t)}{dt}$. Compute the derivative of $x(t)$ and then substitute $t = 3$.

• (c) Acceleration: The acceleration is the second derivative of the position function, $a(t) = \frac{d^2x(t)}{dt^2}$.

2) Analyzing velocity and acceleration from a graph

The second problem gives a velocity vs. time graph with two points: $Q$ and $P$. You're asked to determine:

• (a) Acceleration of $Q$: Acceleration is the slope of the velocity-time graph. Calculate the slope between the origin and $Q$.

• (b) Acceleration of $P$: Find the slope of the line from the origin to $P$.

• (c) Time when $P$ and $Q$ meet: You need to determine when both lines have the same velocity.

• (d) Distance from the initial position when $P$ and $Q$ meet: This can be found using the area under the velocity-time graph.

• (e) Velocity when $P$ and $Q$ meet: This is the velocity where the two lines intersect.

3) Projectile motion

The third problem is a projectile motion question, where a bullet is fired at a target 91.4 meters away with an initial velocity $v_0 = 427 \, \text{m/s}$.

• (a) Find two possible angles ($\theta_1$ and $\theta_2$): Use the given identity for projectile motion and solve for the angles that satisfy the range equation.

• (b) Time of flight: Determine the time required for the bullet to reach the target for both angles.

Would you like me to provide detailed solutions for any specific part of this problem set?