The velocity of a shred traveling in the y direction follows V_y = 3t² m/s, and the time is in seconds. At time t₀ = 0 s, the starting position of the shred is y₀ = 2 m. Calculate the position of the particle at t = 1 s.

The equation v_x² = (nP₀/m)t describes the velocity of an engine-powered device for time t₁ = 0 and t₂ = 30 s.  During this time interval, the engine is at its peak acceleration. The equation contains an engine's power in multiples of P₀ (nP₀) in watts, the device's mass, m in kg, and v in m/s. The equation is interpreted as v² having a direct relationship with t. Derive an expression for the acceleration in terms of P₀, m, and t that can be used to determine acceleration at a time, t ≤ 30 s.

The velocity of a ball left in the wind is described by the expression v_x = 3ct² m/s; time is measured in seconds, c is a constant. At t₀ = 0 s, the ball has an initial position x₀ = - 8.2 m. At a later time t₁ = 4.1 s, the ball is located at x₁ = 8.2 m. What is the value of the constant expressed with the right units?

A dog has escaped from its cage and has run 8.2 km away towards a park. The dog has returned to its cage after it has escaped. The dog has taken 25 minutes to run from the park back to the cage. If the origin is positioned at the cage and the dog has run towards the +x-axis, what was the dog's average velocity in m/s for its return from park to the cage?

Over the course of a year, a space station travels approximately 10¹² km as it orbits the Earth. What is the average speed of the space station in kilometers per hour (km/h) during its journey around the planet?

Two bikes are to reach a destination simultaneously. Both start at the same time and are at an equal distance from the destination. They both travel in straight lines toward it. Bike P has variable average velocities: 30 km/h for the first 80 km, 45 km/h for the next 50 km, and then back to 30 km/h for the remaining 100 km. The bike T maintains a constant velocity throughout the journey. Determine the speed of the bike T to ensure that both bikes arrive at the same time.

A school bus is heading to school at a constant velocity. After covering a distance of 140 m in 5.0 s, the driver applies brakes on a traffic signal and comes to rest in 4.0 s. Determine the magnitude of its acceleration in i) m/s² and ii) in terms of the gravitational constant (g).

The following graph represents the position of a vehicle on a straight road with respect to time. Determine the vehicle's instantaneous velocity at t = 15 seconds.

Consider a vehicle's motion along a linear path, and the graph below illustrates its position as a function of time. Determine the instants if any where the velocity of the object becomes zero.

A video is a collection of motion pictures measured in frames/second. Ordinary cameras can attain 60 frames/second. Cameras are used to capture and analyze movements occurring very fast, typically to the order of milliseconds. The nearly vertical launch of a 450 g and 42.0 cm long projectile is shown in the graph below. Use the graph to determine the acceleration of the projectile at 10 ms, 25 ms, and 40 ms. Hint: Approximate the curve to a straight line.

The figure below depicts a velocity-time graph for the motion of a truck. Determine the average acceleration of the truck between i) t = 5.0 s and 30 s ii) t = 0.0 s and t = 25 s.

The velocity versus time data of a remote control car collected during a high school experiment is shown in the figure below. The car moves along a straight line. Initially, at t=0 s the car was located at x₀= 1.0 m.  Find the position, velocity, and acceleration of the car at t = 40 s.

Leaps are very common among animals. A scientist uses high-motion pictures (videos) at 3000 frames/second to analyze the motion of a frog. Frogs have the ability to leap about 20-50 times their length. The motion of a 24 g and 2.52-inch long frog at a nearly vertical take-off angle is plotted using data from motion pictures. Use the graph to determine the maximum height reached by the frog in the first 75 ms and 225 ms.

Find the total displacement of an object during the time interval from t₁ = 2.0 seconds to t₂ = 4.5 seconds if the velocity function is v(t) = 30 + 12t, with v in m/s and t in seconds. Make use of calculus.