Velocity-time graph sketching

4. A car is moving along a straight horizontal road. The speed of the car as it passes the point \(A\) is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the car maintains this speed for 30 s . The car then decelerates uniformly to a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is then maintained until the car passes the point \(B\). The time taken to travel from \(A\) to \(B\) is 90 s and \(A B = 1410 \mathrm {~m}\).

1. Sketch, in the space below, a speed-time graph to show the motion of the car from \(A\) to \(B\).
2. Calculate the deceleration of the car as it decelerates from \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Question 4 continued \(\_\_\_\_\)

9 A car travelling in a straight line accelerates uniformly from rest to \(V \mathrm {~ms} ^ { - 1 }\) in \(T \mathrm {~s}\). It then slows down uniformly, coming to rest after a further \(2 T\) s.

1. Sketch a velocity-time graph for the car. The acceleration in the first stage of the motion is \(2.5 \mathrm {~ms} ^ { - 2 }\) and the total distance travelled is 240 m .
2. Calculate the values of \(V\) and \(T\).

5 A child is running up and down a path. A simplified model of the child's motion is as follows:

• he first runs north for 5 s at \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\);
• he then suddenly stops and waits for 8 s ;
• finally he runs in the opposite direction for 7 s at \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Taking north to be the positive direction, sketch a velocity-time graph for this model of the child's motion.

Using this model,

calculate the total distance travelled by the child,

find his final displacement from his original position.

3 A cyclist starts from rest and takes 10 seconds to accelerate at a constant rate up to a speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After travelling at this speed for 20 seconds, the cyclist then decelerates to rest at a constant rate over the next 5 seconds.

1. Sketch a velocity-time graph for the motion.
2. Calculate the distance travelled by the cyclist.

A man drives a car on a horizontal straight road. At \(t = 0\), where the time \(t\) is in seconds, the car runs out of petrol. At this instant the car is moving at \(12\) m s\(^{-1}\). The car decelerates uniformly, coming to rest when \(t = 8\). The man then walks back along the road at \(0.7\) m s\(^{-1}\) until he reaches a petrol station a distance of \(420\) m from his car. After his arrival at the petrol station it takes him \(250\) s to obtain a can of petrol. He is then given a lift back to his car on a motorcycle. The motorcycle starts from rest and accelerates uniformly until its speed is \(20\) m s\(^{-1}\); it then decelerates uniformly, coming to rest at the stationary car at time \(t = T\).

1. Sketch the shape of the \((t, v)\) graph for the man for \(0 \leq t \leq T\). [Your sketch need not be drawn to scale; numerical values need not be shown.] [5]
2. Find the deceleration of the car for \(0 < t < 8\). [2]
3. Find the value of \(T\). [4]

The driver of a car accelerating uniformly from rest sees an obstruction. She brakes immediately bringing the car to rest with constant deceleration at a distance of \(6\) m from its starting point. The car travels in a straight line and is in motion for \(3\) seconds.

1. Sketch the \((t, v)\) graph for the car's motion. [2]
2. Calculate the maximum speed of the car during its motion. [3]
3. Hence, given that the acceleration of the car is \(2.4\) m s\(^{-2}\), calculate its deceleration. [4]

A cyclist starts from rest and takes 10 seconds to accelerate at a constant rate up to a speed of 15 m s\(^{-1}\). After travelling at this speed for 20 seconds, the cyclist then decelerates to rest at a constant rate over the next 5 seconds.

1. Sketch a velocity-time graph for the motion. [3]
2. Calculate the distance travelled by the cyclist. [3]

A cyclist starts from rest and takes 10 seconds to accelerate at a constant rate up to a speed of \(15\text{ m s}^{-1}\). After travelling at this speed for 20 seconds, the cyclist then decelerates to rest at a constant rate over the next 5 seconds.

1. Sketch a velocity-time graph for the motion. [3]
2. Calculate the distance travelled by the cyclist. [3]

A cyclist starting from rest accelerates uniformly at \(1.5 \text{ m s}^{-2}\) for \(4\) s and then travels at constant speed.

1. Sketch a velocity-time graph to represent the first \(10\) seconds of the cyclist's motion. [2]
2. Calculate the distance travelled by the cyclist in the first \(10\) seconds. [2]